Content uploaded by Johannes Reich

Author content

All content in this area was uploaded by Johannes Reich on Sep 23, 2023

Content may be subject to copyright.

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 dierent economic subjects.

Methodology: I proceed in several steps. First I provide a graph-based

model for the innitely 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 dierence between her valuation and her costs; (ii)

seller's utility depends only on the dierence between his revenue and his

cost; (iii) the total utility generated by the trade is the dierence 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 signicantly 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 dened 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 'unscientic'. 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

'scientic'

] 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 dierent 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 dierent persons have dierent prefer-

ences (reected in, say, dierent 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 dened 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 innitely 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 dierent, 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 fulll 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 dierence between

her valuation and her costs, (2) seller's utility depends only on the dierence

between his revenue and his cost, (3) the total utility generated by the trade

is the dierence 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-

eciency 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 briey 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 dierence 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 innitely 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 innitely 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 dierent ways, namely exter-

nally and internally [33], as I illustrate in Fig. 2. The external coupling of

complementary roles of dierent 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 dierent 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 dierent ways roles can compose correspond to two dierent

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 dierence 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 dierent 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 dierent roles of the same subject, their coordina-

tion, is a restriction of the product transition system of the dierent 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 indierent

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 reects 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 dierent 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 dierent 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 innite items on each level.

As is well known, a utility function over-species 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-specication, 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 dened 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 articial extension of our preference relation?

7

Because we intuitively, that is without reection, 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 dierent pairs of these state

values by her preference relation: namely she is indierent per denition 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 dene a utility function, where

B

gains from a trade interaction

the dierence 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 dierent (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 indierent 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 dene the utility of

S

, he gains from a trade interaction as the

dierence 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

'prot' 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 dierent. For example, we could have dened

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 benet 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 fullled 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 dened 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 dierence 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 denitions 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 dierently in every subject by its neurons and

perhaps accessible by introspection, but as a concept to simplify our reasoning

about subjective preferences under well dened 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 fulll our three trade-constraints as well as the trader constraint.

First, I dene:

Denition 1.

A money based trade between a buyer

B

and a seller

S

is a

mechanism which fullls the following constraints:

Trade-interaction constraints:

(a) The utility of

B

depends only on the dierence 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 dierence 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 dierence 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=fx+y

2

(9)

for all

x, y ≥0

.

I rst show the linearity for rational coecients 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=

f2nz+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

fz

n=f(z)

n

for all

z≥0

and

n= 1,2, . . .

.

Together we then have for the rational coecient

q=m

n

with

m, n =

1,2,3, . . .

:

f(qz) = qf (z)

.

To extend this result to general, real coecients 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 coecients.

Be

a=Piαipi

,

b=Piβipi

,

c=Piγipi

three elements of

R

with the sets

of rational coecients

{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 indierent 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 inuence 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 dened 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 dierence 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 dene 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 dene a relation

A

prefers

g

more than

B

that is satised if and only

if

valA

g> valB

g

So, its the fulllment 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 fullled, 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-ecient'

, 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-eciency 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-inecient', because at least the super rich would be

worse o. So what?

Likewise, it is also well known that 'Pareto-eciency' 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 dierent 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 dierently.

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 dened by

denition 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 dierent interactions and not the apples.

7.3 Wellfare economics

Welfare economics is an area of economics that suered particular from the

avoidance to intersubjectively compare material preferences. It is simply not

possible to do welfare economics without weighing the welfare of dierent 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

dened, 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 inuence 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 dierent 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 fullled, money can aect us very positively by supporting

our social coordination ability very much and thereby increasing our autonomy.

But if not, it can also aect 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

inuence 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 dierence 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 eects.

Instead, usually, starting with an ordinary trade, spending money also aects

the preferences of others. So, as it is wise to endow dierent people in their

political roles with dierent political power, because people dier 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 conict, 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 denition, 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 aairs 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 dierent 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 dierent 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