Investor Presentaiton
Exposition (1738, reprinted Econometrica 1954)
THE MEASUREMENT OF RISK
25
that a rich prisoner who possesses two thousand ducats but needs two thousand
ducats more to repurchase his freedom, will place a higher value on a gain of
two thousand ducats than does another man who has less money than he.
Though innumerable examples of this kind may be constructed, they repre-
sent exceedingly rare exceptions. We shall, therefore, do better to consider
what usually happens, and in order to perceive the problem more correctly
we shall assume that there is an imperceptibly small growth in the individ-
ual's wealth which proceeds continuously by infinitesimal increments. Now
it is highly probable that any increase in wealth, no matter how insignificant,
will always result in an increase in utility which is inversely proportionate to the
quantity of goods already possessed. To explain this hypothesis it is necessary to
define what is meant by the quantity of goods. By this expression I mean to con-
note food, clothing, all things which add to the conveniences of life, and even
to luxury anything that can contribute to the adequate satisfaction of any
sort of want. There is then nobody who can be said to possess nothing at all in
this sense unless he starves to death. For the great majority the most valuable
portion of their possessions so defined will consist in their productive capacity,
this term being taken to include even the beggar's talent: a man who is able to
acquire ten ducats yearly by begging will scarcely be willing to accept a sum of
fifty ducats on condition that he henceforth refrain from begging or otherwise
trying to earn money. For he would have to live on this amount, and after he
had spent it his existence must also come to an end. I doubt whether even those
who do not possess a farthing and are burdened with financial obligations would
be willing to free themselves of their debts or even to accept a still greater gift
on such a condition. But if the beggar were to refuse such a contract unless
immediately paid no less than one hundred ducats and the man pressed by credi-
tors similarly demanded one thousand ducats, we might say that the former is
possessed of wealth worth one hundred, and the latter of one thousand ducats,
though in common parlance the former owns nothing and the latter less than
nothing.
$6. Having stated this definition, I return to the statement made in the pre-
vious paragraph which maintained that, in the absence of the unusual, the utility
resulting from any small increase in wealth will be inversely proportionate to the
quantity of goods previously possessed. Considering the nature of man, it seems to
me that the foregoing hypothesis is apt to be valid for many people to whom this
sort of comparison can be applied. Only a few do not spend their entire yearly
incomes. But, if among these, one has a fortune worth a hundred thousand ducats
and another a fortune worth the same number of semi-ducats and if the former
receives from it a yearly income of five thousand ducats while the latter obtains
the same number of semi-ducats it is quite clear that to the former a ducat has
exactly the same significance as a semi-ducat to the latter, and that, therefore,
the gain of one ducat will have to the former no higher value than the gain of a
semi-ducat to the latter. Accordingly, if each makes a gain of one ducat the
latter receives twice as much utility from it, having been enriched by two semi-
ducats. This argument applies to many other cases which, therefore, need not
26
DANIEL BERNOULLI
be discussed separately. The proposition is all the more valid for the majority
of men who possess no fortune apart from their working capacity which is their
only source of livelihood. True, there are men to whom one ducat means more
than many ducats do to others who are less rich but more generous than they.
But since we shall now concern ourselves only with one individual (in different
states of affluence) distinctions of this sort do not concern us. The man who is
emotionally less affected by a gain will support a loss with greater patience.
Since, however, in special cases things can conceivably occur otherwise, I shall
first deal with the most general case and then develop our special hypothesis in
order thereby to satisfy everyone.
N
B.
R
$7. Therefore, let AB represent the quantity of goods initially possessed.
Then after extending AB, a curve BGLS must be constructed, whose ordinates
CG, DH, EL, FM, etc., designate utilities corresponding to the abscissas BC,
BD, BE, BF, etc., designating gains in wealth. Further, let m, n, p, q, etc., be
the numbers which indicate the number of ways in which gains in wealth BC,
BD, BE, BF [misprinted in the original as CFI, etc., can occur. Then (in accord
with $4) the moral expectation of the risky proposition referred to is given by:
PO m.CG+.DH+ p.EL + g.FM + ...
Now, if we erect AQ perpendicular to AR, and on it measure off AN - PO, the
straight line NO-AB represents the gain which may properly be expected, or
the value of the risky proposition in question. If we wish, further, to know howView entire presentation