M.A.
ECONOMICS
Albert B. Goracz
EXPECTED UTILITY,RISK,AND THE THEORY OF ECONOHIC CHOICES.
FIRST COPY.
CONTENTS
I. Introduction
i.
A. MATHEMATICAL PROBABILITY APPROACH
II. Prelimimary Remarks on Utility
III. The Derivation of Equi-Expectancy Curvee
IV. The Cardinal Measurability of Utility
1.
6.
12.
V. The Principle of Diminishing Marginal
Utility of Money
VI. The Friedman-Savage Theory of Games
VII. Empirical Verifications'
18.
20.
26.
B. SUBJECTIVE RISK/FEELING APPROACH
VIII. The Difference letween bubjective and
Mathematical Probability
27.
C. CONCLUSIONS
31.
D. BIBLIOGRAPHY
32.
,
- l -
INTRODUCTION.
This thesis deals with the formation of consumption
decisions in the presence of uncertain alternatives and
intends to be an axiomatic evaluation of the li ne of thought
originated, or rather renewed by Neumann and Morgenstern.(#)
The utility analysis of economic choices has offered,
besides throwing some light on the process of choosing,
two still controversial results :
i/ Utility is numerically measurable; and
11/ The marginal utility of money has increasing values
over certain intervals.
It does not seem to be necessary to stress the
significance of the first result.
If it is true then the
postulate of ordinal measurability of utility, which has
lead only to a scientific nihilism mainly in connection
with the so called "welfare economics", can be discarded.
The whole system rests on seven particular assumptions,
three of which will be deduced directly from readily
acceptable axiome.
Although it i8 possible to trace back
the remaining four assumptions to statements ofaxiomatic
character and then show that the system ofaxioms is insufficient, for the sake of demonstration, l thought it
best to determ1ne their degree of plausability.
Beside
these assumptions, stated in the text, the validity of
innumerable axioms is presupposed.
The first of them is
that uncertain alternatives exist even in connection with
·.
the consumption function.
Risk factors entering the in-
veetment and production functions can be taken into account
by using the principle of profit maximisation and without
referring to such concepts as utility. (##)
WeNhave to
mention ônly the results of household budget surveys to
prove that propositions (such as Little's
~
certainty can be neglected in connection with
choices cannot be maintained.
) that unconsume~
There is hardly any
situation in which consumers are not faced with alternatives
to which some degree of risk is attached.
The results of the utility analysis of "uncertain
Il
choices is not restricted to the field of economics, there
are mathematical and linguistic achievements which are
worth noting. The difference between subjective and
probabilityhas lead to the revision of the orthodox mathematical
"objective" (mathematicaïlrPFObability theory and some
other aspects of the theory of games has persuaded some
of the authors of the necessity of working out an entirely
new branch of mathematics.
From the linguistic point of
view such new expressions have occured as "expected clarifying of expectations" or "potential surprise".
In the literature on this subject we find several
rather startling statements.
Samuelson declares that the
whole question belongs to the field of aesthetics, or,
at least, it is only its aesthetic aspects he is interested
h~
in andlProposes that the so called "strong independence
(#) J. Neumann &. O. Morgenstern: Theory of Games , f-,;/IIf(,l.fo-lA f I~"""
See e.g. Kalecki: ~~~ Increasing Risk,Economica,1937
(HI)
Studies in Economie Dynamics,19 43.
.. ,.
,
-z.iz..
axiom" should be tested in an octane-rating engine.
style used by the writers varies from
....
~
The
the most formal
and rigourous treatment (as e.g. certain parts of the reply
written by Friedman and Savage to Baumol) to story and
even joke-telling.
There is nothing fundamentally original in this study
and it seems to be rather unlikely that anybody could give
reliable answers to the questions stated without an
extensive statistical investigation and a complete revision
of the probability theory. . Nevertheless, the proposition
that the expected utility function contains an element of
gain or loss arising from the inevitable change in the
disutility of labour function is - as far as l know - new
and so are the critical remarks made without reference.
In the text frequent use will be made of mathematical
symbols but no particular knowledge
of' mathematical technique is presupposed.
- 1 -
PRELIMINARY REMARKS ON UTILITY.
It follows from the axiom of consistent human behaviour
that an individual, when confronted with two or more alternatives, attaches or behaves as if he attached a homogeneous
quantity, called utility, to each alternative and chooses the
one with the greatest utility.
Utility in this formal sense
is simply an ordering quantity and in Most cases concerning
economic behaviour it is rather irrelevant to what it
actually means, pleasure or displeasure, enjoyment or pain,
desire or dislike.
Usually it is sufficient to postulate
the existence of some quantity which the individual tries to
maximise when making an economic choice.
The utility concept has been built up entirely upon
"
introspection and was worked out by using primitive, dinner
1.
table analysis.
Consequently, the term "utility" meant for
economists a simple, homogeneous, sensual-emotional phenomenon:
the direct, physical enjoyment derived fram using a particular
commodity.
It i8 8ufficient to refer to two facts in order to make
clear that physical enjoyment or pleasure has no direct influence upon economic choice.
Jl!'11xX'Jdrtxprmm=~"XXJf1m~;r~~.
The firet fact is that there is a time lag between choice and
physical enjoyment: an individual cannot derive any physical
pleasure from using a commodity before he has it.
It follows
that the utility we have to consider in this theory exists
- 2 -
only in the imagination of the individual, it ia an
expectation,(an ,ex ante) phenomenon.
vidual's utility
expec~ations
Further, the indi-
can be determined by past
experiences only. J J:ep?-~..:'....L
eKcmple utility is a logical
evaluation of past sensual-emotional feelings, - if we
assume that individuals take into consideration physical
enjoyments only.
The second fact is the infinitesimal
character of the physical pleasure impulses.
This proves,
beyond doubt, the logical character of utility.
The physical pleasure impulses are originated by small
particles of an object consumed, therefore, there is no
such sensual-emotional phenomenon, as for example, pleasure
derived from consuming one pound of bread.
When consuming
one pound of bread the individual in question experiences
innumberable pleasure impulses, which are supposed to be
unaddible, but the utility of one pound of bread as a whole
is a logical evaluation, aggregated from these infinitesimal
pleasure feelings.
If thia is true, the argument of non-
measureability of utility loses much
o~
its convincing force.
Although the statement that human feelings cannot be measured
numerically is quite plausible, it is not so obvious why
logical phenomena are numerically unmeasurable.
The separation of sensual-emotional and logical processes
in connection with utility has been longago made by Pareto.(#)
(#)
,
1 Id
1
Nous etudierons les actions logique, repetees, en grande nombre,
qu'exe~utent les hommes pour se procurer les choses quisatisfont
(Cont'd)
- 3 -
But Pareto assumes a strong correlation between these two
phenomena.
The Paretian assumption is obviously static
in character.
In a dynamic system, a correspondence of
past experiences and expectations i8 untenable, but even
in its static form it is doubtful since it bas been pointed
out that there are other elements in the utility concept
beside the logical evaluation of physical pleasures or
enjoyments.
It would be an interesting psychological study
to evaluate the determinants of the individuals' utility
feelings but, as we tried to show, from an economic point of
view the concrete meaning of utility is rather irrevelant,
and, therefore we can restrict ourselves to the three
results of the above argument :
i/ Utility exists and is an ordering quantity;
ii/ Utility is a logical rather than a sensual-emotional
phenomenon;
iii/ Utility is an ex ante phenomenon, it belongs to the
field of expectations.
The first statement can be directly deduced from the
axiom of consistent human behaviour, the second and the
third results fOllow from the fact that there is a time lag
(H) Cont'd
~
~
leur gouts. Nous simplifierons encore le probleme, en supposant
que le fai\jpouvons le,faire parc~ue nous ne considerons que
des actions qui se répetent ce qui nous permet d'admettre que
c'est un lieu -log~que qui unit ses actions. qR ho~e qui pour
la première fois achète ijn certain aliment, pOfra acheter plus
qu'il ne lui en faut pour satisfaire ses gou~s ••• mais a qp
seconde achat il rectifiera son erreur ••• et ainsi, petit a
petit il finira par . se procurer ce dont il a besoin ••• s'il
se trompe une premi~re fois dans ses rais9ngement au sujet de
ce. q~~il de~ir:S-l les rect~fiera enJ;;~.cr:~e~~:~./~~,.145,146.
f-u.k1tC--ht;: ç ~O<4pk- rcvtv.-,~~ ~ ~f- ~t;.~ ,."f ~
- 4 between the process of using and that of choosing an economic
object and from the fact that emotional impulses are
infinitesimal in character.
The two facts referred to in
the preceeding sentence are ofaxiomatic character: the
truth of them is obvious and it is impossible to imagine,
even theoretieally, a situation in which they are not valide
EXPECTED UTILITY.
If we denote by~he probability that an economic event
will occur, then (1-00 may be called the risk factor attached
to every alternative open to the individual.
The value of
(l~) may vary fram 0 to l, the former value representing
the impossibility of the occurence of the event in question,
the latter the certainty of occurence.
From this point of
view economic events can be classified into three groups:
i/ Events connected with a risk factor " (l~) •
o.
(Certain events)
ii/ Events connected with a risk factor 1> (1- (i»
o.
(Uncertain events)
The third group,
(1-0(> = l
has no practical importance
since impossible events can have no utility, they cannot be
chosen.
Expected utility means the utility of an uncertain event
which ean be deduced from the utility of "certain events".
If we denote by U(A) the utility of an economic object, A,
(with 0 risk factor
att~ched),
then the expected utility,
- 5 -
Ue' oE an event involving the object A and a risk Eactor
(1-
o() can be written :
= E [U(A); (1- rAÙ
II.1.
where 1) (1- o()
0
Under this heading time preEerence can be explained, since
there is always some time lag involved between the time
point oE expectation and the time point of outcome of a
risky event.Taking this factor into consideration, our
general Eormula should be written
II.2.
Ue
= E [U(A);(l-oO;t]
where t is the amount of
time between the two points
involved.
In Most cases conneeted with consumers' choices the time
element is not very important, and consequently, we shall
neglect it.
It 1s quite obvious, on the other hand, that
the omission of the time factor in connection with the investment function may lead to serious misrepresentation.
As we have shown, the utility of "certain" events is
expectational in character and so is "expected utility".
Therefore utility is homogeneous in both cases (logical, ex
ante phenomenon); there is no qualitative difference between
the utility of certain events and the utility of uncertain
events.
- 6 -
THE DERIVATION OF EQUI-EXPECTANCY CURVES.(#)
We have defined the expected utility of an uncertain event
as a function of the uti1ity of the object in question and
a risk factor:
III.l.
Ue
=
f [(U(A) ; (14
D
where A is the object which may rea1ize
and (l-~ ) is the risk factor attached
to it.
From this general form two rather trivial results can be
deduceda
i/ If in two uncertain events the objects involved are
identical, or the individual is indifferent between them, he
will choose the alternative to which a smaller risk factor is
attached; and
ii/ If in two uncertain events the risk coefficients are
equa1, the individual will choose the alternative whose object
has greater uti1ity.
In the theory of probability mathematical expectation is
defined as the product of the amount an individual may win
or lose and the probability of winning orlosing.
For
examp1e if one has a 0.5 chance of winning $1,000. his
mathematica1 expectation is #500.00.(##) By using similar
arguments, we may write our general function in the following
specified form :
(#) The term is taken over from Davis: Theory of Econometries,
the idea it covers comes from Bernouilli.
~
{##} In most cases there are usually more expectational elements
involved, one hardly can expect to win anything without the danger
of losing something. The complete mathematical expectation
includes the whole range of all the possible winnings and losses.
- 7 -
= U(A) ,d...
111.2.
where U(A) i5 the utility of object
A when it is certain and 0( is the
probability that it will realise.
If we assume that different objects (with 0 risk coefficient or with l probability coefficients) can be expressed
in terms of money our formula May be transformed into :
111.3.
Ue
= U(Ma),o(
where Ma is the money equivalent of
object A and 0( is a probability coefficient.
Further, by assuming that the marginal utility of
money is constant over the whole range considered, we may
write :
U
e
=).. {Ma> 0(
'f
where ~ is a constant
(#)
and using the constant ~ as unity of measurement of utility,
the equation becomes :
Ue
=
Mao<.
which is the formula of mathematical (money)
expectation.
Under these assumptions the mathematical money
expectation and utility expectation numerically eoincide.
Taking more than one event into consideration, we shall
arrive at the commonly used formula of mathematical expectation :
(#) It should be mentioned here the difficulty of giving any
meaning to absolute zero utility.
We shall follow the prac-
tice (originated by Bernouilli) of eliminating the difficulties
- g -
III.6.
where Mi is the amount of'
money (or utili ty) and
0(If,
i8
the probability attached to it.
If' we keep Ue constant and change the values of' Mi andoti!
we obtain a set of' equivalent propositions among which the
individual must be indif'f'erent, provided that our assumptions
hold true.
Following this method we may draw a set of' indif'f'erence
curves, called equi-expectancy curves, i.e. loci along
which expected utility is constant.
E.g. a proposition of'
winning $1,000.00 with 0.1 probability ls equivalent to the
proposition of' winning $10,000.00 with a 0.01 probability.
Further, since utility is expectational in character, equiexpectancy curves and indif'ference curves (in the ordinary
sense) are homogeneous, so that risky events can be related
to certain events, and we may postulate that individuals
have a complete ordering of' alternatives.
Cont'd.
connected with the question of absolute zero utility by
(#)
assuming that the origin of measurement of utility is not
absolute zero.
We find very convincing the explanation of
Bernouilli that there is some level of wealth below which
individuals cease to behave rationally.
This "wolf-point"
may wall be taken as the origin of integration of the marginal
utility of money function.
- 9 In connection with the so called
St~ Pete~~bourg
paradox
'0).1
~ eur:>~h()...".pl~~ ~#VI.1I/f.A~ M/Ht-t"1 o}/Wf.~ ()u4"'~h~
Bernouilli had to removeyfor it his much discussed (and
denied) proposition that the satisfaction which a man has
in adding an increment of wealth (dU) to his fortune of x
unit is directly proportional to dx and inverselyproportional to x :
dU
where dU i8 the increment in
utili ty and
l' is a constant.
From this equàtion Bernouilli derived the formula of moral
expectation (#) :
III.B.
Ue" y
.
0<
Ki +f'] -
l
where
0( is
the probability of
winning amount Ma and y is the
individual's wealth.
Using the formula for moral expectations WB May obtain
by the above described method equi-expectancy curves and
surfaces.
The following diagram, taken over from Davis,
shows a set of equi-expectancy curves under the assumption
that Ue
10, and y is $100.00, $1,000.00 and c;:..6
lf"" ..
IQo
(0
t'
.,,0
)0
'tl1
lb
1.
(#) The mathematical derivation of the
f~la
of moral
expectation is too wall known to reproduce it here.
See: Davis
:~.
The Theory of Econometries.
Bloomington,1941 •
,
,
'
- 10 -
Obviously, it is very questionable that the shape of the
curve representing the marginal utility of money can be
described by a simple function as the Bernouilli postualte.
R. Frish and Jordan, to mention only two, tried to evaluate
the shape of the marginal utility of money curve from statistical data but the statistical proof given by them is
not very convincing.
In more recent studies, as we sball
see when discussing the analysis of Friedman and Savage,
even the constantly diminishing character of the marginal
utility of money is denied.
The above argument is based, among others, on four often
doubted assumptions:
,
i/
ii/
iii/
utility is cardiaally
~easurable;(~)
utilities are independent;
thè:',èxpectedrutility of an uncertain event can be
derived from the utility of the object in question (with
zero risk factor or with l probability) and no other elements
are to be considered;
iV/
the subjective risk felt by individuals can be
correctly expressed by objective mathematical probability.
The rest of this the sis is devoted to the examination of
these assumptions.
The firet and the second assumptions
will be discussed in conneetion with the analysie of Neuman
and Morgenstern, the thlrd when dealingwith the propositions
- Il -
of Friedman and Savage and the fourth in connection with
Shackle's system.
(#)
The proposition that the marginal utility of money is
constant, diminishing or
assumption.
increas~ng
necessitates this
The ordinal measurability of utility does not
reveal anything of the increasing, diminishing or constant
charaeter of the marginal utility of money.
be <l'.uite obvious and can easily be verified.
This seems to
For a
rigourous mathematical proof see Samuelson : Foundations of
Economics·'~~~f /t(~f.
.Iv.- THE CARDINAL MEASUREMENT OF UTILITY.
- 12 -
The Neuman-Morgenstern proposition can briefly be stated as
follows : The numerical measurability of utility can be derived
from the ordinal measurability of utility if we assume that
consumer units are able to tell their ordinal preferences
not only between objects (with 0 risk factor or 1 probability
coefficient) but a1so between two risky alternatives or between a certain object and" a risky event. (#) This additional
assumption i8 equivalentto saying that the consumer units'
ordinal preference ordering is complete (it includes uncertain
events) and consistent.
The actual derivation of the possibility of cardinal measurement
of utility from the ordinal preference relations or consumer
units is quite simple :
Let C, A and B be certain events (objects with 1 probability
coefficient) for which the consumer unit's
one stated.
preference is the
If we introduce a fourth event, D, which consists
of a (l-c() chance of getting B and an ~chance of getting C, or:
IV.l •
D i'
(1- o()B
t o( C
,
. then it is possible to change the value of 0< until the consumer
unit reaches an indifferent position between D and A, i.e.
IV.2.
U(D)
=
U(A)
(#) The treatment by indifference curves implies either too much
or too little: if the preferences of the individual are not aIl
comparablethen the indifference curves do not existe If the
individualts preferences are all comparable, then we can obtain
a numerical utility which renders the indirference curves superfluous. "Neuman - Morgenstern: Theory of Games and Economie Behaviour, Princeton, 1944, p.20. And (in the usual indifference
curve ana1ysis we expect an individual) "for any two alternative
events which are put before him to be able to tell which of the
two he prefers. It is a natural extension of this picture to
permit such an individua1 to compare not on1y events but even
combinations of avents with stated probabi1ities. NeumanMorgenstern, op.cit.p.13.
- 13 -
Since A is preferred to B and the outcomê of D is either B or
C, it follows that, when confronted with a choice between A
and D, no matter which of
the ~ alternatives
it chooses, the
consumer unit will get at least an amount of utility equal
to the utility of B. Consequently, it is not necessary for
the consumer unit to measure total utility (from the absolute
zero point), it is sufficient if it measures the difference in
utility between A and B on the one hand and between C and D
on the other.(#)
Putting the utility of B equal to zero
U(D)
becomes ~. U( C).
IV.3.
U(D)
= o(U(C)
Substituting IV.3. into IV.2. we May rewrite the Indifference
situation expressed by IV.2. :
IV.4.U(A)
=0(
U(C)
which is equivalent to
IV. 5 • U( A) - U(B )
= of
E(
C) - U( B
J
since U(B) is zero.
We should like to emphasize that in the last three equations
utilities are maasured from the utility of B (which is therefore
equal to zero) and consequently U(A) means the difference in
utility between A and Band, similarly, U(C) is the difference
in utility between C and B.
Since the probability coetttëient
ci is a real number between zero and one, it May be taken as a
numerical estimate for the ratio of the preference of A over B
to that of C over B. (##)
(#) Introspection tells us that this i8 the actual mental process
of the individusl in such a situation.
(##) "Consider three avents, C, A, B, for which the order of the
individual's preferences is the one ststed. Leto(be a raal number
between 0 and l, such that A is equally desirable with the com-
- 14 E.g. if the consumer unit declares that it is indifferent
between A and D when
between A and B 1s
!
0(
is ! then the .utility difference
of the difference 1n uti11ty between
C and B, or, measur1ng
~tility
from Band choos1ng U(C)-U(B)
as the unit y of measurement, the utility of A is!.
Th1s
method can be eas1ly extended, for example, the numerical
uti11ty of E 1s
~
if the consumer unit feels to be in an
1ndifferent position between E and
18
[j 1- o() B +d.. C]
when
\X
!, provided tbat the ord1nal preference system of the
consumer un1t 1s C, E, B, and the orig1n of measurement 1s
U(B) and the unity ia chosen as U(C)-U(B), U(B) being zero.
Therefore ut1l1ties are measurable except for orig1n and
unity.
f}L~~
(Neuman and Morgenstern saw tbat the above argument loses
much of its clar1ty and perhaps its convincing force if we
assume that the mere act of gambling has sorne utility.
If the utility of gambling 1s not zero the expected utility
of an uncertain event can be written as :
IV.B.
where Ug ls the utillty of gamb11ng
wh1ch becomes zero when 0( = l,
Ma is the money equivalent of the
object which may realise, and 0(
is a probability coefficient.
and the proof of cardinal mea surabi lity of utility reduces to
~h~ is~i!l
(##) Cont'd
plausible) proposition that utilities are numerically
bined event ~onsist1ng of a chance of probability (l~) for Band
the remaining chance of probabilityO(for C. Then we suggest the
use of ~ as a numerical estimate for the ratio of the preference
of A over B to that of C over B. "Neumann-Morgenstern,op.cit.p.
- 15 measurab1e because there is a real number in the expected
utility fUnction.
It is rather probable that there are other elements in the
expected utility function, besides the utility of gambling,
which will not be zero when
ott
1.
In connection with the
analysis of Friedman and Savage we shall prove the existence
of another utility element which will not be zero when ;fl,
the gain in utility arising from the inevitable change in the
disutility of labour.
The second criticism raised by Samuelson and proved formally
by Malinvaud, (#) is that the Neumann-Morgenstern analysis presupposes that utilities are independent and therefore the system
rests upon two assumptions :
Assumption 1.
The consumer unit's ordering is complete and consistent;
Assumption II. (Strong independence axiom) If lottery ticket
Al is as good or better than Bl' and lottery
ticket A2 1s as good or better than B2 then
an even chance of getting Al or A is as good
2
or better than an even chance of getting Bl
or B
2
(##)
(#) Ma11nvaud: Note on Neumann-Morgenstern!s strong independence
axiome Econometrica, 1952.
(##) In the literature on thia subject
we sometimes find more
assumptiona mentioned such as: If the object A is preferred to the
object Band B to the object C, there will be some probability
combination of A and C, such that the individual is indifferent
between it and B. Such an assumption i8 superfluous since it can
be derived from the basic axiom that the expected utility of an
object A i8 increasing or decreasing as the probability coefficient
attached to it increases or decreases. This axiom holds true on1y
in the original Neumann-Morgenstern system where the utility of
gambling ia postulated to be zero.
- 16 -
There can be 1itt1e doubt that utilities areinterdependent
and therefore the reliabi1ity of Assumption II. depends upon
the degree of a dependence of utilities, the order of magnitude
of which is unknown, a1though perhaps it cou1d be determined
by careful evaluation of household budgets.
~'"
To avoid the difficulty of dealing with the uncoRfortable
question of the utility of gambling, Samuelson assumes that
the strong independence axiom implies gambling: prises are so
determined as to include the utility of gambling.
It would be
interesting to know how, sinee this proposition presupposes
an expected utility function of the following form :
IV.9.
Ue
:rJ,
(Ml
+ Ug ) -+ cXz. (M2 + Ug )
•••••
where Ml' M2 , •••••• , ~ are the possible
outcomes of one choice, Ug is the utility
of gambling and eX110( 1- ). ~ '.
)0(. -{
are pro-
bability coefficients.
It seems to be unnecessary to prove that equation IV.9. does
not correspond to the actual mental process of choosing and
therefore, using Samuelson's term, the proposition ia nonoperational, not to mention that equation IV.9. is equivalent
to the statement that individuals distribute their utility
of gambling by applying the probability coefficients of the
possible outcomes as weights of distribution; and, since any
distribution of any quantity must be numerical in character,
the Samuelson proposition is usel8Bs in a Neumann-Morgenstern
type of analysis because it presupposes the numerical
- 17 -
measurability of utility (i.e. that of gambling) before it
bas proved it.
The actual mental process is determining the
expected utility of an uncertain event can be described as
follows :
IV.10.
Ue = Ug-+2.·MiC<~
where the signs Mean the same as in IV.9.
In terms of our four assumptions (p.15) the Neumann-Morgenstern analysis can be described as an attempt made to prove
the first assumption by presupposing the validity of the other
three.
- 18 -
V. THE PRINCIPLE OF DIMINISHING MARGINAL UTILITY OF MONEY
AND THE THEORY OF THE GAMES.
Let us suppose that an individual has n units of money.
If the principle of diminishing marginal utility of money holds
true, then the utility derived from the nth unit must be greater
than the utili ty of the (n
U{~~ U(~11')
+1) st unit,
or
If in such circumstances an individual is offered a "fair"
game, i.e. he has even chances of winning or losing the same
amount, he has to refuse it since "the gain in utility from
winning a dollar will be less than the utility from losing a
dollar, so that the expected utility from participation in the
game is negative." {#}
But people not only engage in fair games, they engage freely
in such unfair games as e.g. lottery.
Marshall tried to resolve this contradiction by rejecting
utility maximisation as an explanation of choices involving
risk.
Gambling, he claims, involves an economic loss even when
conducted on fair and even terms.
People gamble only because
they overestimate their objective chances of winning, or, in
other words, gambling is an example of irrational behaviour.
This argument according to Marshall requires no further
assumption than that "firstly the pleasures of gambling may be
neglected and, secondly f'(x) is negative for all values of x
where f(x} is the pleasure derived fram wealth equal to x." (##)
(#) Friedman-Savage: Utility analysis, J.P.E.
Ilqh!.
(##) Marshall: Principles, Mathematical Appendix,IX.
- 19 -
There are other possible interpretations of the fact that
people engage in fair (and in unfair) games.
One line of
thought, suggested by Friedman and Savage postulates
the
strict validity of the axiom of rational human behaviour and
neglects the utility of garnbling, from which it follows that
the marginal utility of money is not constantly diminishing
but over some intervals it is increasing.
Another possible
explanation consists of the proposition that the utility of
gambling cannot be neglected and, since we have no information concerning its order of magnitude, the increasing,
constant or diminishing character cannot be deterrnined fram
the analysis of fair games.
Further it can be denied that
the axioms and postulates used in the ab ove argument are
sufficient; e.g., it May be argued that a low correlation
between the individuals' subjective risk feeling and the
"objective" mathematical probability is not a sign of irrational behaviour.
In the following pages we shall examine this argument and
(as far as it is possible) the degree of plausability of the
assumptions upon which the above outlined solutions reste
- 20 -
VI. THE FRIEDMAN-SAVAGE THEORY OF GAMES.
(~
The proposition of Friedman and Savage can briefly be
stated as follows :
If the Neuman-Morgenstern original assumption concerning
human behaviour are sufficient and true, and the following
five observations of facts are correct then it may be proved
that the curve representing the marginal utility of money is
not continuously diminishing but it has increasing segments.
The five facts that the Friedman Savage analysis tries to
"rationalile" are :
1.
..2.
7,
Consumer units prefer larger to smaller incomes;
Low-income consumer units buy or are willing to buy
insurance;
3.
Low-income consumer units buy or are willing to buy
lottery tickets;
4. Many low-income consumer units buy or are willing to
buy both insurance and lottery tickets;
5. Lotteries have more than one prize.
For simplicity, let us regard the alternatives open to the
consumer unit as capable of being expressed in money terms,(##)
or money incomes.
Let us suppose, further, that a consumer
unit May choose between two alternatives, A and & Alternative
A
•
h
cons~sts
0 f(lM.
• 0(canee
0f
•
gett~ng
•
l 1 and a (l-,~
N)
an ~ncome
chance of getting an income 1 , or :
2
A .:l(I l
(1- {)( )12
where 12> Il
Alternative B consists of a certain income 10.
-
-t
(##) Friedman and Savage: "Income is to be understood in the
widest sense embracing, e.g. cash income, schedule of cash income
over time, baskets of goods, fate in love and war, etc." .~ 2 0.- ~~
Friedman and Savage: ~ Mi4r~ Cl,uat;/JS ft du7~ 4,M~W,Ay Jù:.',. O'/lt./7.I".
. .
.
- 21 The expected uti1ity of A may be written :
VI.l.
U(A)
= J U(Il)
+ (1-0( )U( 1 2 )
The actuarial value (mathematical expectation) of A is
VI.2.
I(A)
If
=r/.
I(A)
=
Il
+
(1-0( )1 2
the game is said to be fair.
If under
this condition the consumer unit chooses A, its act must be interpreted that the expected utility of A is greater than the
utility of a certain income of the amount of I(A), i.e.
VI.J.
U(A»
U[I(A}
There must exist a certain income, 1 4, which has the same
utility as alternative A:
If we take into account that utility increases with income
we may say that inequality VI.J. implies
VI.5.
14> I(A)
and the consumer unit would be willing to
paya maximum of 14 - I(A) for the privilege of gambling.
Similarly, if l4 ("I(A), the consumer unit would be willing to
pay a maximum amount of I(A)-I 4 for insurance.
We have pointed out earlier that if the marginal utility of
money is constant over the whole range considered then mathematical expectation and expected utility numerically coincide
(provided that the unit of measurement of utility is chosen
to be the utility of one monetary unit) and individuals should
be indifferent between the actuarial value of an uncertain
- 21 !he Ixp.ct.d utility ot A ..y be written 1
VI.l.
U(A)
= o(U(I l ) 1- (1- o<')U(I 2 )
The actuarial value (math.matic.l expeotation) 01 A 1.
I(A).
VI.2.
+
ci Il
(1- 0( )Ia
It
I(A)
•
la
the
sam.
i. ..id to be taire
It und.r
thi. condition th. consumer unlt choc ••• At ita Act mU8t be int.rpreted that
expected utillty ot A 18 ,reat.r th&n th.
th~
uti11ty ot a e.rtain income
VI.3.
or
th. amount
or I(A),
i •••
U(A)~ U(IA)
Th.r. mu.t ex1.t • c.rta1n incoa.,
I~,
whieh ha. th. aame
uti11ty •• alternatiy. AI
Il
lN
ft
tak. into acceunt that utility inere.... vith ino_.
11&1 My that in.qua1ity VI.,_ iap11 ••
VI.S.
14> I(A)
.nd th. cen.um.r un1t weuld be willlns to
or IIt. - I(A) tor th. privi1.p of pmblinc.
Simil&r1y, il Ï. ( ·I(A), th. CODIUIl.r unit weuld b. willilll to
4
pay • _ximURl
pay a maximum &Mount ot I(A)-I 4 tor in.urane ••
v. haY. polnt.d out .arli.r tbat ·it th. maraina1 uti1ity ot
mon.y i. Gonatant oy.r the Whol. rance con.id.red then math.matieal .xpectation and expected utility numerically coinclde
(provld.d that th. unit
to be the
utili~y
or ••..ur••ent or
utility ia cho..n
of on. mon.tary unit) and indiYiduala ahould
be inditt.rent bet...n th. actuarial y.lue ot an unc.rtaln
- 22 -
income and a certain 1ncome of the same amount.
Consequently,
the only possible interpretation of 1nequal1ty VI.J., in the
presence of the assumpt10ns stated, is that the marginal
uti11ty of money function must have 1ncreas1ng values over
the interval
I(A)~I2.
The following example will clar1fy these concepts:
An 1ndividual 1s offered a choice between two alternatives,
A and B.
Alternative A consists of a
~
chance of getting
$500 and of a ! chance of getting $1,000 therefore its
actuarial value can be written as:
VI.6.
I(A)
= !
of $500 - ! of $1,000
= $750
If alternative B consists of a certain income of the
amount of $750, then the game is said to be fair.
the utility of $500 as origin
o~
Choosing
measurement, the expected
utility of A and the utility of B are:
VI.7.
U(A)
=!
[Ù(500)-U(5OO)]
= !U(500)
U(B) = U(750)
- U(500)
-1- !&(l,OOO) - U(500] =
•
U(250)
If the marginal utility of money is constant over the
interval $500~$1,OOO (i.e. U(M 50l) = U(M 502 ) • •••• • U(Ml,OOO)
then we may use the (constant) utility of one monetary unit as
the unit of measurement of utili ty
VI.à.
U(A)
=
~500
U(B) = 250
t
(r )
and we may rewrite VI. 7. :
= 250 ~
P
In these eircumstances the individual must be indifferent between the two alternat1ves.
If, on the other band, 1ndiv1duals
- 23 -
show by their market behaviour that they prefer A to B, this
means that
U(A)
>
U(B), or
!-500~ .> 250 cp
which is possible only if the average utility
of money is greater between $750-7 1,000 than that between
.500~750,
or, in other words if the marginal utility of money
function has increasing values in the interval $750. $1,000 •
~
In the presence of the assumptions that incomes and probability
alone determine the preference of a given person{u e =2U{Mi)<X'.,;,]
the results of the Friedman - Savage analysis are "true", although the >term "marginal utility of money" should be given
not the usual, but a rather broader meaning.
But since the
result that the marginal utility of money is increasing over
certain intervals was obtained by bringing in connection a
theoretical system containing assumptions of various degree
of abstraction with an empirical, "real" system where those
assumptions do not hold true, the reliability of the propositions
depends on the degree
o~
abstraction
o~
the assumptions.
I.e.,
the proposition that the marginal utility of money is increasing is true only if the removal of apy of the assumptions
would not possibly change the results achieved with the original
system ofaxioms and postulates.
But if we remove the
assumption that the utility of gambling is zero, the expected
utility function becomes Ue • Ug+[U(Mi)
that the difference Ir-I(A)
and it may be argued
is nothing else but the effect of
- 24 -
the utility
o~
gambling.
It can be se en that the question
whether or not the marginal utility
o~
o~
money is increasing
depends on the actual magnitude of Ug , i.e. :
if
Ug~ Ir -I(A)
Ug
the marginal utility of money is increasing;
>
Ug
Ir -I(A)
= Ir
""
-I(A)
""
-
"""
i5 decreasing;
"""
is constant.
\
__7
It is not easy,~rovided that Ir~I(A)J to determine the
order of magnitude
o~
the utility of gambling.
In the
literature on this subject we may find impulsive and highly
romantic remarks such as: "Most human motions tend •••• to
assimilate themselves to the game spirit"(X) or, "what men
want i5 not so much to get
.-e
lW'
things(Chey want as it is to
have interesting experiences." (XI)
It seems to be enough to
refer without a thorough statistical proof-to the low rate of
labour migration to prove that such statements, exciting as
they are, have no serious foundation and that the habit persistence of the large mass of people is rather high.
On the
other hand people engage in games whose "unfairness" is so
considerable that it can hardly be explained by the utility
o~
gambling.
The term "marginal utility of money" should be interpreted
rather broadly.
The usual meaning of the expression refers to
(x) Knight,F. : Risk, Uncertainty and Profit.New York,l92l.p.53
(XX) Op.cit., p.54.
- 24fA -
the purchasing power of money the rate of which must be decreasing if the assumption of perfect divisibility holds and
there is no "increasing return to scale" in consumption. (X)
When dealing with uncertain choices, the utility of money
cannot be restricted to purchasing power, it necessarily includes
a gain or a loss arising from the inevitable change in the
disutility of labour function.
If an individual is offered a
game, he will take into consideration not only the purchasing
power of money but also the fact that he may, by accepting the
game, increase his income without offering more labour, the
utility of which is as far as low-income consumer units are
concerned, very likely negative.
The total utility of money schedule in this sense consists of two components: the utility of goods and services
an individual can purchase with the amount in question and
the utility of labour.
In the equilibrium position the
marginal utility of money in this broader sense is zero since
the individual equates the positive utility of the purchasing
power of money to the negative utility of labour.
Let us suppose that an individual has the following
marginal utility of money schedule (using the positive utility
of.leasure sacrificed in the place of the negative utility of
labour) :
(X) There is one more point here. The validity of the Paretian
assumption that ex ante utilities are equal to ex post utilities is
very questionable in connection with uncertain choices. It is quite
possible that a low-income consumer unit when offered a game in
which it may win a large amount of money - having no previous
experience-overestimates the utility of the sum the game promises.
- 25 -
al
bl
unit of money
1.
utility of goods
and services
purchased
50
2.
3.
4.
TOTAL
4
45
40
35
170
cl utility of
,leasure (#)
10
15
22
35
$2
Marginal utility
of money (b-c)
40
30
1$
0
$$
individualts~orizon
If the
of expectancy" (a term borrowed
from Tinbergen) is 10,000 days and if we assume, for simp1icity,
that every parameter invo1ved remains constant, he expects an
income of the amount of $40,000 or a total of $$0,000 "uti1s".
Imagine now that he is offered a fair game, e.g., he has a
1
tÔ!666
40,000 chance of getting $40,000 and a
losing $1. The expected uti1ity of the game is :
1
=
40,000
e
U
1
(170+ $2)10,000
chance of
-~.35
= 2$
since the game offers not only purchasing power but "free"
purchasing power, i.e., it makes it possible to acquire money
without giving up any uti1ity of leisure.
The 35 utils in
equation VI.9. represent the marginal utility of 1eisure
sacrificed in earning the last unit of money.
The
individu~l
ex/Olt
cannot risk the purchasing power of (more correctly therlltility
of goods and services bought by) his last dollar because he
does not have it.
In the above exarnp1e two alternatives are
open to the individual to choose from: he may gamble and he may
buy goods and services.
If he enters the game he may expect
2$ utils; if he does not (i.e., he buys goods and services) he
will end up with zero utility.
Therefore in this case the fair
game is more profitable.
r#T Saërificed in acquiring the unit of money in questions.
- 26 -
VII.
EMPIRICAL VERIFICATIONS.
An experiment conducted in the Laboratory
o~
Social
Relations at Harvard University by Mosteller and Nogee (#)
tried to verify the propositions of Friedman and Savage by
letting 17 participants play a fairly complicated game.
Without going into details we reproduce here the results of
the experiment:
1. "There is some support
o~fered
~or
the inflection-point analysis
by Friedman and Savage, although this support
is not wholly satisfactory.
There is no contradiction
but the support is meager.
2.
Subjects are not so consistent about preference and
indi~ference
as postulated by Neumann and Morgenstern."
From a theoretical point
o~
view, the main objection
against the experiment is that, although it was based explicitly on the Neumann-Morgenstern axioms, implicitly it was
assumed that the individuals t subjective risk feeling and the
objective mathematical probability are not strongly correlated.
The participants, while playing, had in front
o~
them a list
containing their "true" mathematical probabilities and therefore their attention was concentrated upon the risk
~actor,
and, since the game was conducted ratber fast, they did not
have much time to evaluate their marginal utility of money
schedule.
the
ft
It is surprising that in such circumstances any of
- 27 -
VIII. THE DIFFERENCE BETWEEN SUBJECTIVE AND MATHEMATICAL
PROBABILITIES.
The main points of criticism raised against the use of
mathematical probability
coefficien~
in the theory of choices
involving risk are the following :
1.
Individuals are interested in, and influenced by,
not only the probability of an event but also the askewness
of the probability distribution. (#)
2.
Individuals' subjective feelings that an event will
occur cannot be numerically measured; subjective probability
is only ordinally measurable. W)
3.
Human feelings cannot be
(oIa-u1(,)
added. (###)
4. The mathematical probability theory deals with repeated
trials while economic choices are usually not repeated in a
sufficiently large number.
5.
(###)
Subjective probability is not as detailed as the
objective mathematical probability, e.g., an individual can
hardly feel any difference between, let us say,
a
1
500,000
and a
1
450,000
probability.
(###)
- 28 -
Although unable to offer any objective proof, and referring
only to common sense, l would venture the statement that
in forming their degree of belief that an event will occur,
~(Uf ~
Ot
individuals regard the mathematical probability coefficient
~,4.t~
as the predominant factor.f If we denote the degree of sub-
'D~(j( )
jective probability bY~ we may say that
•
f ( i ,x,y,z, ••••••• }
whereo( is the mathematical probability
coefficient and x,y,z, ••• are the
other factors,
and in a certain range, e.g., betweena
most powerful element.
=l
-;>
l~O )
0(
is the
It seems to .be very likely that over
this critical value, at least as far as games of chance are
considered,~ is
simply a function of the prizes offered.
The only complete system based on subjective probability
coefficient is that of Shackle.
Utility in his definition is
"desire" which is considered to be a sensation.
Since to each
possible outcome there is a risk factor attached, its desirableness is lessened.
Consequently the utility (which means desire
in this case), D, of an uncertain outcome of an event is a
function of the desirableness of the object in question, D(A)
and the individual's subjective degree of belief;1 that an
event will realise, or :
VII.l.
Da = F[D(A);I](#}
(#) "The entity which gives us enjoyment by anticipation has two
sets of characteristics. The first set specifies or de scribes the
situation •••• saying what it would be like if it were to happen
(without saying anything as to whether it will happen). The second
set •••• consists merely in our degree of belieL" Shackle :
Expectation in Economics. p.ll.
- 29 -
Since the degree of belief is a logical phenomenon, it
has to be replaced by the corresponding sensation "surprise".
By doing so a system is created in which only sensations and
emotions existe
The idea of surprise, or rather potential surprise because
we deal with expectations, lies between zero and that intensity which would arise from the occurence of an event
believed impossible, and is supposed to be subject to manipulation by the method of differential calculus.
According to Shackle, in the actual process, individuals
determine D for every possible out come of an event and then,
aince human feelings cannot be added, concentrate their
attention on the best and worst possible outcomes (focusloci).
In other words, the utility function VIII.l. has
two extreme values and only these are taken into consideration.
We should like to make only two remarks in connection with
Shackle's focus-loci theory.
Firstly, in forming their degree
of potential surprise, individuals necessarily take into
account, to sorne extent, the existence of the other outcomes
because even subjective degrees of belief add up to 1.
Secondly, the focus-loci theory contradicts reality, namely
the fact that lotteries have more than one prize.
Let us
,~
suppose that a lottery has four prizes : $100.009, $50,000.00
and $40,000.00
to which equal potential surprise
va,)
is
attached, and $1.00 (which the individual May lose) with ~~
potential surprise.
Then, if the focus-loci theory is true,
- 30 -
the individual will concentrate his attention on two values :
$100,000.00 and $1.00.
By reducing the number of prizes to
two: $190,000.00 and $1.00, the individual should be in a preferred
position (unless we make sorne forced assumption concerning the
formation of the degree of potential surprise.) which contradicts to the facts.
The proposition that the subjective degree of belief is
a function of the prizes offered (over a certain level of~' )
is rather plausible in connection with the mathematical
probability approach which permits the addition of the expected
utilities of the possible outcomes, but it is not very convincing in connection with a focus-loci theory.
People may
not be so logical as Neumann and Morgenstern suppose them
to be but they are far more logical than to deserve to be
forced into so narrow a system as Shacklets.
- 31 -
IX. CONCLUSIONS.
This study wishes to support four propositions :
1.
Utility is a logical and ex ante phenomenon;
2.
The expected utility function contains a gain or
loss element arising from the inevitable change in the
disutility of labour lunction;
3.
It cannot be proved by the analysis of "uncertain"
choices that the marginal utility of money schedule has
increasing values over certain intervalsj
4. The locus-loci theory of Shackle contradicts to
the lacts.
•
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tt
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~'B'arY1rJiè.
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%Dr
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