Thinking and Deciding

254 NORMATIVE THEORY OF CHOICE UNDER UNCERTAINTY
utility” (in the terms of Kahneman and Snell, 1992) may obey different rules than
true utility, a point we shall return to later.
When individuals make decisions concerning amounts of money that are very
small relative to their total lifetime income, they will do best in the long run if they
essentially ignore their risk aversion and think about expected value. For example,
suppose you have a choice of two automobile insurance policies. One policy costs
$100 more per year than the other, but the less expensive policy requires that you
pay the first $200 of any claim. If your probability of making a claim in a given
year is .4, your choice each year is between a sure loss of $100 and a .4 probability
of a $200 loss, an expected loss of .4 · $200, or $80. Over many years, you will
save an average of $20 per year by purchasing the less expensive policy, even though
it involves a greater risk. (The same reasoning applies to maintenance contracts.)
Because your savings account will have $20 more in it each year (or you will owe
your creditors $20 less), your savings will add together over time. The declining
marginal utility of money is more relevant when we are considering large amounts
of money accumulated over a lifetime.
The fact that the marginal utility of money is generally declining has implications
for the distribution of income and wealth in society. If $1,000 means more to
me when I have only $2,000 in my bank account than when I have $200,000, then it
seems reasonable to assume that, in general, $1,000 would mean more to poor people
in general than to rich people in general. Accordingly, in designing a system of
taxation, it makes sense to require the rich to pay more. Such an uneven distribution
of the monetary burden makes the distribution of the utility burden more even.
It also allows the government to impose the smallest total utility burden across all
taxpayers in return for the money it gets. In fact, most systems of taxation operate on
some version of this principle. Similar arguments have been made for directly redistributing
wealth or income from the rich to the poor. We must, however, consider the
effects of any scheme of redistribution or unequal taxation on incentive to work. In
principle, it is possible to find the amount of redistribution that will strike the ideal
compromise: In this state, any increase in redistribution would reduce total utility by
reducing incentive, and any decrease in redistribution would reduce total utility by
taking more utility from the poor than it gives to the rich in return.
Exercises on expected-utility theory 7
Assume that someone’s utility of receiving an amount of cash X is X .5 .
1. Make a graph of this function. (It need not be exact.)
2. What is the utility of $0, $5, and $10?
3. What is the expected utility of a .5 chance of winning $10?
4. Compare the expected utility of the gamble with the utility of $5. Which is greater?
5. Compare the expected value of $5 with the expected value of the gamble.
6. Compare the expected utility of $5 with the expected utility of the gamble if the utility of
X were X 2 instead of X .5 .
7 Answers to selected problems are found at the end of the chapter.