Money as a mechanism to intersubjectively compare entirely subjective preferences

Abstract

Purpose: This article contributes a money model that let us understand under which conditions money allows to interpersonally compare the entirely subjective material preferences of different economic subjects. \\[-0.3cm]

Methodology: I proceed in several steps. First I provide a graph-based model for the infinitely scaling network of economic relevant interactions, which resolves the issue that the nodes (economic subjects) and the edges (economic interactions) of this graph cannot be modelled independently. It rests on the role concept, where roles can compose externally to an interaction (an edge) and internally to a coordinating subject (a node).
Second, I choose the preference concept and its possible representation by a utility function, every economist is familiar with, to model how an economic subject takes its decisions.
Then I introduce utility representations for the two roles of buyer and seller, which compose externally to a trade and internally to a trader.
Finally, I prove that under the following assumptions there still exists a (unique) utility representation of the preferences of seller and buyer in a trade: (i) in a trade buyer's utility depends only on the difference between her valuation and her costs; (ii) seller's utility depends only on the difference between his revenue and his cost; (iii) the total utility generated by the trade is the difference between buyer's valuation and seller's cost and is independent of the price; and (iv) the utility of a trader, coordinating both roles, is independent of whether we look at her as seller or buyer.\\[-0.3cm]

Findings: With money interpersonal comparison of subjective material preferences is possible under the following conditions: (1) The decisions of an economic subject in a trade can be explained by a preference model where preferences are expressable as utility, that is they are complete, transitive and non-hierarchical. And (2) all buyers have the same total budget and the same access to the same set of alternative outcomes.\\[-0.3cm]

Value: Since the (now disproved) conviction that one cannot compare subjective preferences interpersonally had significantly shaped economic theory in the past, the author assumes that the (now proven) fact that money allows exactly such a comparison under certain conditions will shape economic theory in the future as well.

Full text

Money as a mechanism to intersubjectively
compare entirely subjective preferences
Johannes Reich, johannes.reich@sophoscape.de
September 23, 2023
Abstract
Purpose: This article contributes a money model that let us under-
stand under which conditions money allows to interpersonally compare
the entirely subjective material preferences of dierent economic subjects.
Methodology: I proceed in several steps. First I provide a graph-based
model for the innitely scaling network of economic relevant interactions,
which resolves the issue that the nodes (economic subjects) and the edges
(economic interactions) of this graph cannot be modelled independently.
It rests on the role concept, where roles can compose externally to an
interaction (an edge) and internally to a coordinating subject (a node).
Second, I choose the preference concept and its possible representation
by a utility function, every economist is familiar with, to model how an
economic subject takes its decisions. Then I introduce utility representa-
tions for the two roles of buyer and seller, which compose externally to a
trade and internally to a trader. Finally, I prove that under the follow-
ing assumptions there still exists a (unique) utility representation of the
preferences of seller and buyer in a trade: (i) in a trade buyer's utility
depends only on the dierence between her valuation and her costs; (ii)
seller's utility depends only on the dierence between his revenue and his
cost; (iii) the total utility generated by the trade is the dierence between
buyer's valuation and seller's cost and is independent of the price; and (iv)
the utility of a trader, coordinating both roles, is independent of whether
we look at her as seller or buyer.
Findings: With money interpersonal comparison of subjective material
preferences is possible under the following conditions: (1) The decisions
of an economic subject in a trade can be explained by a preference model
where preferences are expressable as utility, that is they are complete,
transitive and non-hierarchical. And (2) all buyers have the same total
budget and the same access to the same set of alternative outcomes.
Value: Since the (now disproved) conviction that one cannot compare
subjective preferences interpersonally had signicantly shaped economic
theory in the past, the author assumes that the (now proven) fact that
money allows exactly such a comparison under certain conditions will
shape economic theory in the future as well.
1

1 Introduction
Can we meaningfully compare preferences between economic subjects, despite
the fact that they are entirely subjective? This question is still ercely debated
in modern economics. The contribution of this article is to show that money
can  under well dened circumstances  provide exactly that kind of 'magic':
a mechanism to interpersonally compare entirely subjective preferences.
In fact, in the middle of the 20th century, it was mainstream to think that
interpersonal comparison of preferences was 'unscientic'. To quote some well
known scholars of that time: Lionel Robbins [34], said
'every mind is inscrutable
to every other mind and no common denominator of feeling is possible'
. This
conviction seemed to be the base of his belief that the relation between ethics
[viewed as
'speculative'
] and economics [viewed as
'scientic'
] could only be a
'mere juxtaposition'
[35, p.132]. Kenneth Arrow [1] still said that
'interpersonal
comparison of utilities has no meaning.'
1
.
According to his own account, Amartya Sen [41] was
'much concerned with
incorporating dierent ways of making interpersonal comparisons and their far-
reaching consequences on what is permissible in welfare economics.'
, culminat-
ing in his book 'Collective Choice and Social Welfare' in 1970 [39]. But even
Amartya Sen stated in 1999 [40, p.68]
If dierent persons have dierent prefer-
ences (reected in, say, dierent demand functions), there is obviously no way
of getting interpersonal comparisons from these diverse preferences
If this were true, economics would face a severe dilemma: How should it say
anything sensible about the distribution of scarce but exchangeable goods based
on the preferences of the people, without being able to relate these preferences
to one another?
In sharp contrast to this dismissive attitude, it is well known since many
decades that utility, under well dened circumstances, could even be 'trans-
ferred' from one person to another. This property is the base for game theorists
like John C. Harsanyi [15] (pp. 111) or Roger B. Myerson [24] (pp. 384) to
classify cooperative games into those with and those without transferable util-
ity. The base for this transferability is the representation of utility in a special,
namely a linear function.
To convincingly show that money can be a mechanism that allows for in-
terpersonal comparison of preferences, I have to go through a certain program.
To begin with, I have to nd a way to adequately model the innitely scaling
network of economic relevant interactions, a section of which I show in Fig. 1.
Such a network can be represented mathematically by a graph. But, unfortu-
nately, the edges of this graph, namely the interactions, and the nodes of the
graph, the economic subjects, cannot be modelled independently.
Here, the rst key idea comes into play, which originates from computer
science [32,33], to use a projection of the participants, which I call a 'role'
2
,
1
Economically utility is understood as one form to represent preferences, we come to this
issue later.
2
This role concept was actually very much inspired by a sociology seminar on Erving
Gofman [14] led by Prof. Uta Gerhards at the University of Gieÿen in the 1990s
2

as the common building block for both, the edges as well as the nodes of this
graph.
These roles can compose, or couple, in two dierent, complementary ways,
namely externally and internally [33], as I illustrate in Fig. 2. The external
coupling of these roles result in an interaction, an edge of the graph. In the case
of a buyer and seller role, it results in a trade. Their internal coupling, which
I name
'coordination'
, results in the economic subject, a node of the graph. In
the case of a buyer and seller role, it results in a trader.
The next step in my program is to provide an adequate model of human
mind and action: We act in interaction with many other people in whom we
can empathize. In doing so, we construct our actions by making self-determined
spontaneous and choice decisions according to our preferences under certain
assumptions about the world and our fellow human beings. The problem of how
to construct an appropriate action to achieve our goals under our assumptions
is solved by our intelligence in terms of a problem-solving competence.
In a further step I represent our preferences with a utility function, which
is familiar to every economist. The freedom we have to express our preference
relation with a utility function paves the way to our money model as we can
impose on it several desirable restrictions and can still be sure to talk about our
preferences while instead refering to utility values.
In section
'The trade'
I introduce the money model in an intuitive way by
using the traditional micro economic concept of the utility of a buyer and a seller
role. But beyond traditional economics, I take both, the interaction perspective
as trade as well as the coordination perspective as trader into account.
In the following section I prove that the ad hocly chosen simple linear utility
functions in the former section are indeed the only ones that fulll our intuition
of money as a utility transfer mechanism. They are charcterized by 4 require-
ments, that (1) in a trade buyer's utility depends only on the dierence between
her valuation and her costs, (2) seller's utility depends only on the dierence
between his revenue and his cost, (3) the total utility generated by the trade
is the dierence between buyer's valuation and seller's cost and independent of
the price, and, nally, (4) the utility of a trader, coordinating both roles, is
independent of whether we look at her as seller or buyer.
The remaining step to intersubjectively compare the utility of competing
buyers simply requires all buyers to have the same total budget and the same
access to the same set of alternative outcomes.
'Other approaches'
I present from the literature are: the concept of Pareto-
eciency to avoid intersubjective utility comparison alltogether, the treatment
of money as a commodity, and how welfare economics misinterpreted the utility
concept.
In the concluding
'Discussion'
, I show the relevance ot the presented money
concept by briey addressing several interesting economic questions that it af-
fects directly. Is our model suitable to also adequately describe the trade of our
labor? Is it desireable to endow everybody with the same total budget? Do fair
prices, that partition the utility of a trade equally, guarantee global fairness?
Why is the dierence between a reward and a recognition essential? And -
3

nally I conclude the article acknowleding the deeply rooted connection between
money according to the presented model and our values and culture.
2 A model of the economically acting subject
As said in the introduction, before theorizing about money, we rst have to
model the economically acting subject. The model I present consists of two
interrelated parts, one part is directed towards the economic world, the subjects
acts within and the other part is concerned with how it acts.
2.1 Interactions, coordination and decisions of the eco-
nomic subject
I see the economic world as a virtually innitely scaling network of economically
interacting subjects, a section of which I show in Fig. 1. Thus, I propose to
model this network as a graph, where the nodes are the subjects and the edges
are the interactions.
Figure 1: A section of the innitely scaling network of ecnomic relevant inter-
actions.
This model has to account for the issue that the nodes (economic subjects)
and the edges (economic interactions) of this graph cannot be modelled indepen-
dently: we cannot describe the interactions without refering to the participating
subjects, nor can we describe the subjects and what they do without refering
to their interactions.
The key idea, taken from computer science [32,33], is to use a projection of
the subjects, which I call a 'role'
3
, as the common building block for both, the
edges as well as the nodes of this graph.
3
This role concept was actually very much inspired by a sociology seminar on Erving Go-
man [14] I took part which was sympathetically led by Prof. Uta Gerhards at the University
of Gieÿen in the 1990s
4

In fact, roles can compose, or couple, in two dierent ways, namely exter-
nally and internally [33], as I illustrate in Fig. 2. The external coupling of
complementary roles of dierent subjects result in an
'interaction'
, one or sev-
eral edges of the graph. In the case of a buyer and seller role, it results in a
trade. The internal coupling of dierent roles of the same subject, which I name
'coordination'
, results in the economic subject, a node of the graph. In the case
of a buyer and seller role, it results in a trader.
A)
B)
Figure 2: The two dierent ways roles can compose correspond to two dierent
perspectives. They either compose externally, creating an interaction (part A)
or internally, creating a coordinating subject (part B).
Mathematically, a role can be modelled as a nondeterministic, stateful, dis-
crete input/output transition system whose transition relation is determined by
an additional, complementary input alphabet, its decisions [32]. That is, we
can say that the nondeterminism of our interactions leaves us the latitude to be
lled with our decisions to determine our actions.
Based on the two possible types of nondeterminism, that either a given input
does not determine the transition or a transition occurs without any input,
we can distinguish two types of decisions: selection decisions and spontaneous
decisions. You see the dierence if you ask a child either 'Which instrument do
you want to learn?' or 'Do you want to learn some instrument?'
An interaction is then constituted by the (external) exchange of I/O-characters
between several, complementary roles of dierent subjects. This structure 
without the additional decisions  is well known in computer science as
'proto-
col'
[17]. Together with the decision I call it a
'game in interaction form (GIF)'
[33], showing the tight relation between computer science and game theory in
how they deal with interactions. Its execution is determined by the decisions of
5

the participating subjects.
The internal coupling of dierent roles of the same subject, their coordina-
tion, is a restriction of the product transition system of the dierent roles of a
single subject.
Please note that as I introduced a role only as a projection of a subject
onto its interaction, what seems to be a decision in one interaction may just
be the consequence of the subject being involved in some other interaction.
Coordinating several roles can reduce (up to complete elimination) as well as
increase the number of decisions.
2.2 A simple theory of mind: Preferences
Based on our model of our behavior, we must now provide some model of our
mind that explains how we take our decisions. A quite simple and still expressive
model of our mind, which is used extensively in contemporary economics, is the
preference concept.
With the preference concept, we postulate that within an interaction we act
as we 'want' in the sense that we can 'explain' the decision of a subject to choose
a certain transition because it 'preferred' its outcome over some alternatives.
Given two distinguishable outcomes
a
and
b
, we can then say that we either
prefer
a
over
b
or
b
over
a
, notated as
a≻b
and
b≻a
, or we may be indierent
between
a
and
b
, notated as
a∼b
.
Thereby, the preference concept rests on the simplifying assumption that a
given decision in some interaction always reects what we really want and is
not determined by some other interaction via coordination. Its approximating
character becomes apparent by the fact, that very often, we need to coordinate
our dierent roles to act according to our preferences. For example, we prefer
to have time with our family to do something enjoyable together over working,
but we have to work to earn the money to do so. Thus, in the context of
our preference relation, isolating our dierent roles is only possible to a limited
extent.
In fact, as I will discuss later on, money plays a central role in enabling or
hindering our coordinating capabilities.
2.3 Utility
As is explained in virtually every textbook on economics (for example [22]),
the key idea of the utility concept is to simplify our model of preferences even
further by introducing a single valued function
util
, mapping all outcomes to
real numbers such that we can replace our preference relation by a comparison
of these 'utility' values.
We say that a function
util
represents our preference relation
≿
, if
util(a)≥
util(b)
is equivalent to
a≿b
for every outcome
a
and
b
. We thereby reduce a
property of a binary relation between two outcomes to a property, the utility,
that can be attribute to a single outcome.
6

There are a number of propositions describing under which conditions it
is possible to represent preference relations with utility functions. Above all,
the preference relation necessarily has to be transitive and complete. Mathe-
maticians call such a relation a 'total order'. In economics, such a preference
relation is often termed 'rational'. I think, this is an important misnomer and
it would be much more appropriate to name such a total order 'consistent' from
an economic perspective. Because on the one hand, being rational in colloquial
terms implies much more than having a consistent preference relation and in
real life having consistent preferences is probably rather the exception then the
rule even for decisions, most of us would describe as rational in colloquial terms.
Then, in the countable case with a xed preference relation, there always
exists a representing utility function. Also for continuous preference relations
over convex domains [6]. However, no such representation exist in the important
case of hierarchical preference relations with innite items on each level.
As is well known, a utility function over-species its corresponding preference
relation. If one such utility function
f
exists, then any concatenation
u◦f
with a
strongly monotone function
u:R→R
is also a corresponding utility function to
the same preference relation  and none of these functions is somehow marked.
But beside the issue of over-specication, talking about utility (where this is
allowed per assumption) is really the same as talking about preferences.
'Utility
maximization'
then just means to realize what you prefer most. Thus, the
popular equation of utility maximization with egoism is nonsense, as the utility
concept does not determine the content of the preferences it is supposed to
express. Egoism is much better dened as having preference that are directed
only towards oneself.
To account for uncertain outcomes Johann von Neumann and Oskar Mor-
genstern [43] looked at probability distributions over coutcomes, named them
'lotteries'
, and viewed these lotteries as the to-be-preferred alternatives.
So, in summary, utility is nothing we look for in the brains of people but
is an abstract mathematical concept we use to simplify our considerations on
human preferences where mathematics allows us to do so.
3 The trade: external coupling of buyer and seller
The essential idea of the money mechanism is to extend our preference relation
from single commodities to pairs of commodities and money. I call the prefer-
ences for which this makes sense
'material preferences'
. They are characterized
by a certain exchangeability.
Mathematically speaking, being
M
the set of money values and
C
the set
of commodities, then the preference relation
≿⊆(C×M)2
relates pairs of
commodities and amounts of money. For example, let us assume that we prefer
to have 10
e
and no cinema ticket over having only 5
e
and such a ticket. In
symbols:
(0
cinema ticket
,10e)≻(1
cinema ticket
,5e)
. In this case we would not
pay more then 5
e
for the ticket.
Why should we do such an articial extension of our preference relation?
7

Because we intuitively, that is without reection, grasp the usefulness of this
concept in supporting us in our challenge to coordinate our many social interac-
tions. Part of the money we gain in one interaction we can put on the table in
another interaction. We can now start to realize a lot more actions, according
to our original preference relations, because of this money mechanism  why
shouldn't we prefer something that makes this possible?
So, we can state: money works, because of its capability to support us in
coordinating our diverse economically relevant interactions and nally because
we prefer it to do so as whole subjects.
Next, I apply the introduced role concept to analyze the network of trading
interactions. First, I focus on the interaction perspective of a money-based trade
and secondly, I focus on the complementary coordination or subject perspective.
3.1 Buyer
To describe the money-based trade of a commodity
a
I rst look at a subject in
a buyer role that I name '
B
'. I assume for
B
, that she has a certain amount of
money, her budget,
mB
total
. Her task is now to partition her budget in a way that
she acquires her preferred commodity
a
. That is, to resolve her preference to
acquire
a
she has to partition her budget
mB
total
into her valuation
valB
a
, which
is the maximum amount she would pay to get commodity
a
, and a rest. It holds
mB
total ≥valB
a
.
To express her preference relation, we represent her possession of commodity
a
as a pair
(n, x)
where
n
is the number of commodities of type
a
and
x
is the
amount of money she possess. We now can relate dierent pairs of these state
values by her preference relation: namely she is indierent per denition towards
(0, mB
total)∼(1, mB
total −valB
a)
and  assuming 1 as the unit of money  has
the following preferences:
(0, mB
total)∼(1, mB
total −valB
a)≺. . .
≺(1, mB
total −1) ≺(1, mB
total)≺. . . .
(1)
Thus, we can dene a utility function, where
B
gains from a trade interaction
the dierence between her valuation
valB
a
and her
costB
a
to acquire
a
.
Buyer:
utilB
a=valB
a−costB
a
(2)
We see immediately that for any
costB
a≤valB
a
this function is indeed a
utility function for
B
, as the equivalence
utilB
a(cost1)> utilB
a(cost2)⇔cost1<
cost2⇔(1, mB
total −cost1)≻(1, mB
total −cost2)
holds.
Please note, that the available budget is an essential part of a subject's state
it depends on to coordinate her dierent (economically relevant) interactions.
3.2 Seller
Let's turn to another subject in a seller role which I call '
S
'.
S
has a certain
amount of money
mS
total
before the production of commodity
a
. For produc-
8

tion,
S
had to spend a part of his money, his
costS
a
. With selling he wants at
least to get compensated for his expenses. Hence, he is indierent towards the
following possessions
(1, mS
total −costS
a)∼(0, mS
total)
and the more money he
gets additionally in the trade, the better:
(1, mS
total −costS
a)∼(0, mS
total)≺
(0, mS
total + 1) ≺. . .
(3)
Thus, we can dene the utility of
S
, he gains from a trade interaction as the
dierence between his
revenueS
a
and the
costS
a
he has to produce
a
.
Seller:
utilS
a=revenueS
a−costS
a
(4)
Again, it's easy to verify, that this is indeed a utility function. In economics,
there have been many other terms to name these quantities.
utilS
a
is also named
'prot' or 'supplier rent', while
utilB
a
of the last section is also named 'consumer
rent'.
3.3 The trade
In a trade interaction, the
external
coupling of two subjects in their buyer (
B
)
and seller (
S
) roles is created by the exchange of money and commodity
a
. The
coupling condition is therefore the identity between
B
's cost and
S
's revenue
which we name
pricea
, together with an increment in the quantity of commodity
a
for
B
and a decrement in the quantity of commodity
a
for
S
.
External coupling:
pricea:= costB
a=revenueS
a
(5)
According to our assumption of freedom of choice, both parties will only
agree on this trade if both take at least some utility out of it. Hence,
B
's
valuation as well as
S
's cost determine the possible price range in so far as
S
's
cost determine the lowest and
B
's valuation the highest possible price. Within
this price range, this trade constitutes a 'win-win"-situation for both.
The total utility for both together after trading
a
at
pricea
is just the sum
of the individual utilities and does not depend on the price.
Total utility (trade):
utila,total =utilB
a+utilS
a=valB
a−costS
a
(6)
Please note that this form of total utility depends on our assumptions of the
simple form of Eqs. 2and 4for the utility of
B
and
S
. If we had chosen a
monotonously transformed version of both, then the total utility of trade would
look dierent. For example, we could have dened
utilB
a=log(valB
a−costB
a)
and
utilS
a=log(revenueS
a−costS
a)
. Then total utility would have been
utila,total =
9

log(valB
a−costB
a) + log(revenueS
a−costS
a) = log((valB
a−costB
a)(revenueS
a−
costS
a)) = log((valB
a−pricea)(pricea−costS
a))
, which would not be independent
of the price.
With the assumed utility functions of
B
and
S
, both parties contribute to the
total utility.
B
with a high valuation and
S
with low costs. As total utility does
not depend on the price, we can attribute it to the interaction as an invariant
and say: a free trade creates a utility surplus of x. The price only distributes
the utility between
B
and
S
in the trade  hence, the notion of money as a
mechanism to distribute utility.
I give an example: A book producer and a book enthusiast have an initial
asset of 100
e
each, making a total asset of 200
e
. The book producer now
produces a book which costs him 30
e
, resulting in a remaining asset of 70
e
plus the new book. The book enthusiast would give away 90
e
of his 100
e
to
get this book. Both agree on a price of 50
e
. Subsequently, the book producer
has 120
e
and the book enthusiast has 50
e
and a book he values 90
e
, making
an asset of 140
e
. Together, after the trade, both have a total asset of 260
e
.
The asset grew by the total utility created by the trade of 20
e
for the book
producer and 40
e
for the book enthusiast. So both did benet from the trade,
though not equally.
4 The trader: the internal coupling of Buyer and
Seller
Now we assume that the roles of
B
and
S
are fullled by the same subject, a
trader which I name '
T
', that rst buys the commodity and sells it afterwards.
We thereby create an
inner
coupling of both roles within one subject with the
coupling condition that the commodity has to be bought before it can be sold.
We might think that now, as a commodity is rst bought and sold second,
T
creates utility twice  but this is not the case. Instead,
T
's cost (purchase
price) in her both roles as
B
and
S
are identical. And
T
's revenue as
S
becomes
her valuation as
B
in the dened sense of a maximal amount of money she is
sensibly willing to pay for the commodity. Thereby, also
T
's utility is identical
whether we describe her as
S
or
B
. So, as there is just one person, there is also
only one utility.
Internal coupling:
utilB=utilS
(7)
Then we could take either formula for
B
or
S
to get the total utility of
T
as
the dierence between sale price and purchase price, which obviously depends
on the price(s):
Total utility (trader)
:utiltrader
total =priceS−priceB
(8)
10

5 Money as a utility transfer mechanism
In the last two sections we have seen that our assumption of two simple utility
function denitions for a buyer
B
and a seller
S
in a trade interaction together
with the coupling condition of a 'price', lead to a simple form of total utility,
which was independent of the price. This function of money to work as a utility
transfer mechanism is well known in economics [15,24].
But, as said before, the assumed form of the utility functions is in fact ad
hoc. As we have learned, any combination of a monotone function together with
the assumed utility function would also have resulted in a valid utility function
 but would not have lead to the simple form of total utility, independent of
the price.
Also, the assumption that the total utility generated by a trade is just the
sum of the utility of Buyer and Seller is ad hoc. Again, any combination of an
additional monotone function together with the assumed sum-function would
also have resulted in a valid total utility function.
But we have introduced utility not as something to look for in the brains of
people, which may be encoded dierently in every subject by its neurons and
perhaps accessible by introspection, but as a concept to simplify our reasoning
about subjective preferences under well dened circumstances. Thus, we can
use the freedom of the utility concept not to insist on any particular form, but
to choose exactly that form which makes the interpersonal comparison of utility
meaningful and thereby allows us to compare preferences interpersonally.
I now show that the utility representations I introduced for
S
and
B
are the
only ones that fulll our three trade-constraints as well as the trader constraint.
First, I dene:
Denition 1.
A money based trade between a buyer
B
and a seller
S
is a
mechanism which fullls the following constraints:

Trade-interaction constraints:
(a) The utility of
B
depends only on the dierence between her valuation
and her costs:
utilB
a=f(valS
a−costB
a)
, and becomes zero if her
valuation equals her costs,
f(0) = 0
, and becomes her valuation if the
costs are zero,
f(valS
a) = valS
a
.
(b) The utility of
S
depends only on the dierence between his revenue
and his cost:
utilS
a=g(revenueS
a−costS
a)
, and becomes zero if his
revenue equals his cost,
g(0) = 0
, and becomes his revenue if his costs
are zero,
g(revenueS
a) = revenueS
a
.
(c) The total utility resulting from the trade as the sum of the utility of
S
and
B
is the dierence between
B
's valuation and
S
's cost and
is therefore independent of the price:
utilT otal
a=utilB
a+utilS
a=
valB
a−costS
a
.

Trader constraint: The utility of a trader
T
is independent of whether we
look at her as
S
or
B
.
11

The claim is now,
Theorem 1.
The utility functions of a seller
S
and a buyer
B
in a trade
interaction according to Def. 1, namely
utilS
a
and
utilB
a
is given by equation 2
and 4.
To prove this theorem, we rst have to provide the general utility functions
for
S
and
B
in a trade interaction. As we have seen,
valB
a−costB
a
is a possible
utility function of Buyer and
revenueS
a−costS
a
is a possible utility function of
Seller. Then, the general utility functions for
B
and
S
are provided by two
additional monotonous functions
f, g :R→R
such that
utilB
a=f(valB
a−costB
a)
utilS
a=g(revenueS
a−costS
a)
Now, the trader constraint requires both function to be identical:
f=g
.
And the external coupling condition (Eq. 5) makes the rst constraint to
utilT otal
a=utilB
a+utilS
a
=f(valB
a−pricea) + f(pricea−costS
a)
With the next lemma, we prove that this requires
f
to be linear with zero
intercept.
Lemma 1.
Be
a, b, c ∈R
with
a≥b≥c
and
f(a−b) + f(b−c) = f(a, c)
not
depending on
b
, then
f
is a linear function.
Proof.
4
We can chose
b= (a+c)/2
and get
f(a−b) + f(b−c)=2f((a−c)/2)
.
This value even depends only on
a−c
and not on
a
and
c
individually. As
f(a−b) + f(b−c)
is supposed not to depend on
b
at all, this value holds for
every
b
.
Substitution of
x=a−b
and
y=b−c
results in
f(x) + f(y)
2=fx+y
2
(9)
for all
x, y ≥0
.
I rst show the linearity for rational coecients and in a second step extend
this result to general real numbers.
W.l.o.g. be
f(0) = 0
(The general case with
f(0) = 0
can be reduced to this
one by taking
g(x) = f(x)−f(0)
and looking at
g
instead.)
With induction I prove that
f(nz) = nf (z)
for all
z >= 0
and
n= 0,1,2, . . .
.
The case
n= 1
is trivial. For the induction step, set
x= 2nz
and
y= 2z
in
Eq. (9). Then
(n+ 1)f(z) = nf(z) + f(z) = 2nf(z)+2f(z)
2=f(2nz)+f(2z)
2=
f2nz+2z
2=f((n+ 1)z)
.
4
This proof is due to Martin Härterich
12

By substituting
z
with
z
n
in
f(nz) = nf (z)
, we get
fz
n=f(z)
n
for all
z≥0
and
n= 1,2, . . .
.
Together we then have for the rational coecient
q=m
n
with
m, n =
1,2,3, . . .
:
f(qz) = qf (z)
.
To extend this result to general, real coecients I refer to the fact that
R
is a
Q
-vector space. That is, every real number can be given as a linear combination
of an (uncountable) index-set
{pi}
of real numbers with rational coecients.
Be
a=Piαipi
,
b=Piβipi
,
c=Piγipi
three elements of
R
with the sets
of rational coecients
{αi},{βi},{γi}
. We then have
f(a−b) + f(b−c)
=f(X
i
αipi−X
i
βipi) + f(X
i
βipi−X
i
γipi)
=X
i
f((αi−βi)pi) + X
i
f((βi−γi)pi)
=X
i
(αi−γi)f(pi)
=f(X
i
αipi˘X
i
γipi)
=f(a−c).
What remains is to show that
f(x+y) = f(x) + f(y)
which results from
replacing
x
and
y
in Eq. (9) by
2x
and
2y
.
Thus,
f
is a linear function. To complete the proof of our theorem, we note
that together with our boundary conditions
f(0) = 0
, and
f(valB
a) = valB
a
,
f
has to be the identity function.
We have just proven that to view money as a utility transfer mechanism,
our money based preference relation has to be linear in the money component.
Considering money as a commodity like any other, it was Léon Walras who dis-
covered this peculiar aspect of money and coined the term 'numeraire' [45] for
the money term. Economists traditionally say that our preferences are 'quasi-
linear' with respect to the commodity money, which just means that they are
linear in the money term and arbitrary in any other term.
Two consequences from the linear utility representation of money are well
known (see e.g. [22, p. 45]):
1. if someone is indierent with respect to two bundles
x, y ∈Rn
,
x∼y
, then
any additional amount of money (component 1) does not change that:
(x+αe1)∼(y+αe1),∀α∈R, e1= (1,0, ..., 0)
(10)
2. Money (component 1) is desirable in the sense that everything else con-
stant, we prefer more money over less money:
(x+αe1)≻(x),∀α > 0
(11)
13

This is what our intuition about money is about. If money works as intended,
then, in contrast to ordinary goods, money should have no inuence on our
preference relation of these goods. And, it should always be better to have a bit
more than a bit less money. Actually, these are two well testable propositions
for the validity of our money model.
Please note that in the presented model, even though the money mechanism
can transfer utility in the dened sense, it does not represent or measure utility
in the sense that the more money I have, the more utility I got. So, the statement
'money is transferable utility' is false in this model. True is that money is a
mechanism of utility transfer where individual utility manifests itself on both
sides of a trade only as a dierence between some sort of revenue/valuation and
some sort of cost.
Also the sentence, 'the marginal utility of money decreases with increasing
amount of money' is false in this model. First, we do not attribute utility to
money. And second, within this model utility is a strict linear function of any
money term.
But true is, that our valuation per item of all commodities we buy for con-
sumption, like apples, dramatically decreases with increasing numbers. While
we may be willing to pay 1
e
for a single apple we want to eat now or in a few
days, we will not pay 1000
e
for 1000 apples in a single moment, as we would
perhaps even have to pay additional money to dispose most of them afterwards
somehow.
6 Intersubjective comparison of valuation and util-
ity
If we add a buyer's and a seller's utility to a total utility we obviously relate the
utility of both roles. It allows us to dene a 'fair' price as the price which provides
both the same utility in their trade. But because of their complementarity this
is not what we usually mean by intersubjective comparison  instead we usually
mean the utility of competing roles.
Now, the question is, under which circumstances can we compare the utiliy of
competing roles in a trade, like two potential buyers? The answer is quite simple:
a buyer's contribution to the total utility of a trade is her valuation. We therefore
have to compare the valuation of all buyers. And we can fully understand a
buyer's valuation only if we look at her as a whole, as it certainly depends on
her total budget and also on her set of accessible alternative outcomes in all
of her roles. Thus, comparing utilities of competing roles is only sensible, if
the context of competition allows it, which translates in: all (potential) buyers
must have the same total budget as well as the same access to the same set of
alternative outcomes with respect to themselves. I name this the
'comparable
valuation condition'
.
If two buyers have the same total budget and otherwise have access to the
same alternative actions, then the willingness of one buyer
A
to spend a max-
14

imum of
valA
g
on a good
g
and another buyer
B
to spend a maximum of
valB
g
allows us to draw a direct conclusion about their preferences. Then for
A
(0g, valA
g)A∼(1g,0e)A
holds and for
B(0g, valB
g)B∼(1g,0e)B
holds. We
can thus dene a relation 
A
prefers
g
more than
B
 that is satised if and only
if
valA
g> valB
g
So, its the fulllment of the comparable valuation condition that allows us to
add all utilities of all buyers. Together with our convention, that the total utility
of a single trade is independent of the price, this makes a market mechanism
maximize total utility of all possible trades.
To prove that we just have to order all sellers according to their increasing
cost and all buyers according to their decreasing valuation, and, as Fig. 3
shows, the intersection of both curves determines the market price
p∗
together
with the index
jmax
as this price creates the largest enclosed area and thereby
maximizes total utility.
Figure 3: Supply and demand of a market mechanism.
Without the comparable valuation condition being fullled, we still can add
up all utilities to some value and the market mechanism maximizes this value,
but its relevance remains unclear as its relation to the preferences of the market
participants is lost. So we are not allowed to speak about this value as a 'total
utility'.
7 Other approaches
7.1 Avoiding interpersonal utility comparision: 'Pareto ef-
ciency'
Vilfredo F. Pareto (1848-1923) was the rst to make the distinction between
ordinal and cardinal utility and introduced the idea to handle the analysis of
economic equilibria with ordinal utility [2]. An ordinal interpretation of the
15

utility-numbers allows only to use their ranking while a cardinal interpreta-
tion also allows to use their distance for calculation purposes like equilibrium
determination.
Honouring his achievements, a feasible allocation of resources is nowadays
said to be
'Pareto-ecient'
, if no other such allocation exists, that makes some
consumer better o without making some other consumer worse o [22, p. 313].
Because this criterion is based only on within-subject comparison, applying
it, economists avoid to compare preferences between subjects. In line with
Amartya Sen [41], I think that the wide acceptance of the Pareto criterium
in economics as a model for rational behavior was a direct consequence of the
denial of the possibility to interpersonally compare preferences.
It is well known that Pareto-eciency does not insure an allocation to be
equitable in any sense, but has a strong tendency to justify the distributive
status quo. Just assume we have, by chance, a very unequal wealth distribution,
with a single super rich and a bunch of desperately poor. Any redistribution
of wealth would be 'Pareto-inecient', because at least the super rich would be
worse o.  So what?
Likewise, it is also well known that 'Pareto-eciency' is a bad model for
human decision making which can easily be demonstrated with the example of
the Prisoners' Dilemma [24, p. 98].
Thereby, the denial of the possibility to interpersonally compare utility made
economics a doctrine to justify the economic status quo. It enabled economists
to skip over the question under which conditions the various material preferences
of the dierent economic subjects become comparable. And it removed the issue
of social justice from the economic agenda, culminating in Friedrich Hayek's
strange view of social justice being a
'strictly empty and meaningless'
[16, p.
68] concept.
7.2 Treating money as a commodity
Traditionally, motivated by handling money as any other commodity (see for
example [22,28]), the utility representation of a buyer's preferences is modelled
a bit dierently.
First, it is necessary to set a 'price' for money, which is usually chosen ad
hoc as 1. Then,
B
's utility is not given as the utility exclusively arising from
the trade, but before the trade as its budget
mB
total
and after the trade as its
reduced budget
mB
total
′=mB
total −costB
a
, supplemented by her valuation
valB
a
:
Buyer:
util′B
a=mB
total
′+valB
a.
(12)
This is indeed a utility representation of
B
, as again the equivalence
util′B
a(cost1)>
util′B
a(cost2)⇔cost1< cost2⇔(1, mB
total −cost1)≻(1, mB
total −cost2)
holds. We could have concluded this immediately by recognizing that this
form of utiltiy is the same as eq. 2increased by the initial total budget:
util′B
a=utilB
a+mB
total
. Lets see what the consequences are for the external and
internal coupling of
B
and
S
.
16

For the external coupling of
B
and
S
in a trade, we get the total utility as
before, but with an additional dependency on
B
's initial budget:
util′
a,total =util′B
a+utilS
a
=mB
total +valB
a−costS
a
(13)
However, with this utility representation, the internal coupling of
B
and
S
in a trader does not work, as the internal coupling condition
util′B=utilS
does
not hold any more.
Thus, a model that treats money as an ordinary commodity, except its (quite
extraordinary) linear utility representation, where utility is somehow attributed
to all commodities and therefore also to money, is not compatible with a holistic
view on economic subjects. Such a holistic view, which takes into account all of
the subject's interactions in its roles and the coordination of its roles, prohibits
a utility scale in an absolute sense.
As we have just proven that our utility representation is uniquely dened by
denition 1this is actually no surprise. Se see, a trade is not a symmetric, but
an asymmetric situation with respect to the exchanged items. If we buy a kilo
apple for 2
e
, it's not that the price of the apple is 2
e
per kilo apple and the price
of the money is 2 kilo apple/
e
. Instead the price is part of the external coupling
condition of the trade which consists of exchanging 2 kilo apple at a price of
2
e
and it determines how the total utility of a trade is distributed between a
seller and a buyer. The symmetry breaks down because it's the money we use
to coordinate our dierent interactions and not the apples.
7.3 Wellfare economics
Welfare economics is an area of economics that suered particular from the
avoidance to intersubjectively compare material preferences. It is simply not
possible to do welfare economics without weighing the welfare of dierent sub-
jects against each other. The only question is, what kind of individual welfare
concept to use?
In the tradition of utilitarism, some economists model the individual good
with a unidimensional utility-value [15,39], where every subject
i
draws a well
dened, real-valued 'utility'
Ui(x)
from each state
x
of the world (or as a von
Neumann-Morgenstern lottery). However, as Marc Fleurbaey and Peter J. Ham-
mond [11] indicate, the interpretation of this value is quite heterogeneous. It
could be the extend of pleasure or pain in the tradition of Jeremy Bentham,
happiness, desire satisfaction [4], or, following most economists, preference sat-
isfaction.
I will not go into a detailed critique of these approaches because I think
that there is a fundamental misconception right at the beginning: in our model
total material wealth is given by the sum of all our valuations and not by some
utility. But it makes sense to add up our valuations only if we all have the
same total budget and the same access to the same outcomes with respect to
17

ourselves. Interestingly, valuations are measurable with methods like Vickrey
auctions [42].
The example of the trade between our book producer and our book enthusi-
ast has shown, that a trade between economic subjects increases total wealth by
shifting goods from subjects with lower to subjects with higher valuation and
utility measures (in our model) the extend of the wealth gain and shift between
the trading parties.
But trade is only one out of many mechanisms that inuence our wealth. As
said before, consumption destroys valuation quite naturally. I do not materially
valuate an apple that I have eaten any longer, it's gone. We can also valuate
something we have created by ourselves in the sense that we come to the con-
viction that we will not sell it below some minimum price for whatever reason.
And, there is also bequeath and donation, etc.
And our valuation for one commodity can uctuate enormously, depending
on changing circumstances. Perhaps our book enthusiast, who was just willing
to pay 90
e
for the book, reads the rst few pages and changes her mind, now
thinking: 'Oh what a rubbish! Never ever again should I spend any cent on
this kind of trash!' With this re-valuation she has cut her wealth in one swoop
by 90
e
as she now possess only 50
e
together with a book where she would not
even pay a single cent for, which even may create costs for disposal. Phillip
Nelson [26] introduced the term 'experience good' for goods whose value can be
accurately estimated only after they have been experienced. Or, I have invested
hundreds of Billions
e
in military equipment because I thought I was encircled
by foes, but one day I realize that all of them became friends (although this
seems to be a much better fate then the opposite). Or I just know that other
people only pay 2
e
for this good  why should I pay much more?
In summary the material wealth of a society, as the sum of all its valuations,
is quite dynamic and is governed by sources and sinks. It essentially is a ow
quantity and thereby depends on the relation between its creation and anni-
hilation and the stability of its valuation measures. My presumption is, that
depending on these factors, every society has a certain, characteristic maximal
wealth accumulation capacity. A throwaway society can produce very much
without accumulating valuable goods, whereas a society which heavily invests
in long-valuable goods can accumulate a lot of material wealth even with much
less production capabilities.
But, very important, because of the hierarchical nature of our preferences
and also because there are a lot of things we prefer that we cannot buy directly,
like health, education, freedom, democracy, peace, etc. the sum of all valuations
for our material goods represent societal wealth in a holistic sense only to a
rather limited extend.
8 Discussion
Yes, the money mechanism can achieve the miracle of sensibly relating totally
subjective material preferences of dierent individuals. But it puts strong con-
18

ditions in place to do so: the valued outcomes have to be somehow nonhier-
archically interchangable, everybody has to have the same total budget and
everybody has to have the same access to the same set of alternative outcomes
with respect to herself.
As I have shown, money is not some neutral mechanism that works in the
same way once and for all. But its impact on us, I like to speak of the 'seman-
tics' of money, depends very much on its context of usage. If the mentioned
preconditions are fullled, money can aect us very positively by supporting
our social coordination ability very much and thereby increasing our autonomy.
But if not, it can also aect us very negatively  by the very same function 
being a mean to destroy our freedom of decision and drive us into desolate slav-
ery. Despite the fact that a single free trade is always a win-win situation, with
its unlimited utility transfer function, we can distribute total utility arbitrarily
unfair not only on a local, but on a gigantic global scale. So, to make our world
a better place, it seems essential to understand which handling of money foster
the socially desireable over the undesireable.
To demonstrate the relevance of the presented money model, I outline its
inuence on the treatment of a few other interesting economic issues.
8.1 How do we trade our labor (in a free world)?
Like sugar, as Karl Marx has suggested [21, p. 399]? Then, as a seller of its
labor, that is as an employee, the salary would be the subject's revenue, its cost
would be its expenses for reproduction, and the dierence would be its utility.
Does this makes sense? Compared to selling an apple, which takes place in
a virtual second, our labor time is an essential part of our life and thus it is to
be expected, that in a free world, we would like to express our preference for
some work over others.
Thus, for trading labor in a free world, we have to assume also a discretionary
component on the employee's side, symmetric to that of the buyer, that is the
employer, in the original model. Then the cost term of the employee is no
longer determined exclusively by 'objective' facts, but represent the 'internal
valuation'
5
of the work to be done in the sense of some minimal acceptable
amount of money for which the employee would still opt for doing the job.
The interesting consequence, in case the employee has additional hierarchical
preferences for fairness, is a recursive preference relation with a possible xed
point where the employee's internal and the employer's external valuation be-
come equal, shrinking the formal utility of both to zero [30]. One consequence
is that labor markets cannot work the same way as markets for ordinary goods,
which is already common sense in behavioural economics [8].
Please note, that the role concept also allows to view an organization as a
supersystem created by the interactions of the employees. Then the employees
becomes a part of the organization, which means that there are no interactions
5
The idea to term it that way comes from my wife Christine Reich
19

between the employee and the organization as there cannot be any interactions
between a super- and its subsystems [31].
8.2 Is it desireable to endow everybody with the same
total budget?
Our total budget is a direct measure of our economic power. If we adhere to
the thesis of equal rights and opportunities of all people, then we might think
it a good idea to endow everyone with the same total budget.
However, usually we do not spend our money exclusively for egoistic eects.
Instead, usually, starting with an ordinary trade, spending money also aects
the preferences of others. So, as it is wise to endow dierent people in their
political roles with dierent political power, because people dier enormously
in their capacity and willingness to use their political power in the sense of all,
it suggests, the same holds true for their economic power.
To solve this puzzling conict, the proposal of Richard A. Musgrave [23] be-
comes interesting to divide our preferences into private and public ones (private
and public wants). According to his denition, our private preferences are di-
rected to things that can be acquired exclusively in varying quantities by trade.
Our public preferences, on the other hand, are directed to states of aairs that
apply to everyone equally, non-exclusively, and which are thus beyond simple
acquisition through trade. For example, a person can assert her preference for
an apple exclusively in competition and delimitably against others, while a clean
environment is consumed non-exclusively by all people at the same time.
One possible solution could therefore be to endow everybody with the the
same amount of money for private purposes and quite dierent amounts for
public purposes in Richard A. Musgraves sense. And if someone proves to be
apt to handle public money in the sense of all, it is wiser to provide her or him
more public money as a recognition instead of more private money as a reward
(see below).
In fact, non-exclusivity is a characteristic property of all states to which
successful cooperation refers, if we understand cooperation as an interaction
with a common goal, based on mutual, free agreement.
Thus, we could interpret Richard A. Musgrave's idea to tie the property
of privacy versus publicity to the question of the competitive versus cooper-
ative nature of interactions. Competition requires target states to be taken
exclusively, presupposes similar roles, autonomy of action but only little shared
information, and limits the relationship of competitors to comparatively very
few rules to be followed. Cooperation, on the other hand, requires a common
understanding in the sense of a target state to be jointly adopted and shared
information and allows for a distribution of work between complementary roles.
Obviously, both forms of interaction have very dierent requirements for
distributing resources like information - and money. While for competition to
work out well, equally distributed resources are essential, within a cooperation
it is sensible to distribute resources according to the requirements of the roles.
Already in a trade, only the buyer is in need for money.
20

But looking more closely into a modern society, we see that cooperative
and competitive interaction contexts are quite intermingled, which explains the
complexity of the issue of a sensible money disstribution.
But despite this complexity, we can draw at least two simple conclusions.
The rst is, that any kind of political as well as public economic power should be
tied to a role and not to the subject. As otherwise any attempt to redistribute
it because of a sensible redistribution of roles is very much impeded. Looking
at modern societies, this is already the case to a certain extend. Most money
today is already managed not by any owner but by people in a professional
role. In fact, it seems to me that the extend to which public money is tied to
roles instead of sub