Thinking and Deciding

252 NORMATIVE THEORY OF CHOICE UNDER UNCERTAINTY
Figure 10.1: Utility of a person’s total wealth, according to Bernoulli
per day than with none, and better with two than with one, but if someone is already
giving you five oranges per day, an additional orange would probably make no difference
at all — unless you treated the orange like money and tried to trade it for
something else. Money is less like this than most goods, because it is so versatile.)
Bernoulli suggested that the utility of wealth, for most people, is roughly proportionate
to its logarithm. 6 If this were true, the difference in utility between a total
wealth of 1,000 ducats and a total wealth of 10,000 ducats would be about the same
as the difference between 10,000 and 100,000. The graph in Figure 10.1 shows the
relationship between utility and wealth in Bernoulli’s theory. As shown in the graph,
the value of each additional ducat declines as total wealth increases. Economists
call this the marginal utility of wealth, that is, the utility of wealth at the “margin”
of growth in wealth. The idea of declining marginal utility (with the logarithmic
function) does explain the reluctance of people to spend very much to play the St.
Petersburg game. The extra utility of the high winnings from the very improbable
outcomes (for example, heads on the tenth toss) is no longer high enough to compensate
for their low probability.
The idea of declining marginal utility can explain why people are reluctant to
gamble on even bets. Very few people will accept a bet with “fair” odds. For example,
few people would be willing to give up $10 for a bet in which they win $16 if a
coin comes up heads and $4 if it comes up tails. Of course, $10 is the expected value
of the bet, the average value if the bet were played many times. Figure 16.2 shows
the utility of each outcome of this bet, assuming that the utility of each outcome is
its square root. This function is marginally declining; that is, its slope decreases.
Notice that the utility of $10 is 3.16 (which is √ 10), but the expected utility of the
bet is 3, which is the average of the utilities of its two, equally likely, outcomes (with
utilities of 2 and 4, respectively). The expected utility of the bet (3) is lower than
6 Specifically, Bernoulli argued that “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.” This assumption yields a logarithmic function. Bernoulli gave no other
justification for this function.