lies along a straight-line utility curve. A risk-neutral manager has a linear utility
graph.In fact, the horizontal gap between the CE (read off the curve) and the
expected value (read off the line) exactly measures the discount due to risk
aversion. For any concave curve, it is always true that the CE falls to the left of
(i.e., is lower than) the corresponding expected value.
Figure 12.10 shows three typical utility curves. The concave curve reflects
risk aversion, and the linear graph reflects risk neutrality. The third curve is con-
vex; that is, it becomes steeper and steeper. It is easy to check that an individual
displaying such a curve is risk loving and prefers to bear risk. More precisely, the
individual’s CE for any risk is greater than (lies to the right of) its expected value.
With the utility graph in hand, the decision maker can supply requisite
utility values and routinely evaluate decision trees. Besides assigning utility val-
ues to outcomes, the decision maker can use the graph in reverse. For instance,
the expected utility of the second oil site (56.1) merits drilling. A direct expres-
sion of how much the site is worth to the wildcatter is given by its certainty
equivalent. To find the CE, start at a utility of 56.1 in Figure 12.9, read over to
the utility curve, and then read down to the corresponding monetary value—
in this case, about $50,000. This is the value the wildcatter places on the site.
Thus, he would not sell out if offered $30,000 but would do so readily if offered
a certain $60,000 (or any sum greater than $50,000).528 Chapter 12 Decision Making under UncertaintyCHECK
STATION 4Consider a 50–50 risk between $600,000 and $0. Check that the expected utility of this
risk is 75. Using the utility graph, find the CE of this risk. Compare the risk’s CE and its
expected value. Why is the gap between the two relatively small?Once a utility curve has been assessed, the manager can use the expected-
utility rule repeatedly and routinely to guide his or her decisions. Each particular
decision carries accompanying profits and losses. But what ultimately matters is the
impact of the firm’s many decisions on its monetary wealth position. As a general
rule, it is best to assess a utility function over final monetary wealth. For example,
suppose the wildcatter begins the year with $1.8 million. He thinks about the
potential range of his realized wealthtwo years from now. (This range depends on
the number and riskiness of sites he might explore.) In a worst-case scenario he
might end with debts of $1.5 million. In the best case, his wealth might reach $5
million. Thus, he should assess his utility curve over this wide range.
To sum up, the manager must think hard about tolerance for risks over
different final wealth positions. In doing so, the manager assesses a utility graph
that best represents his or her attitude toward risk.^9 Once the utility curve is in(^9) Decision makers can use a variety of methods to assess utility curves. One such method is pre-
sented in Problem 14 at the end of this chapter. In the process of utility assessment, the manager
can gain considerable insight about his or her risk preferences. For instance, a common finding
is that decision makers become considerably less risk averse when starting from a high (rather
than a low) financial wealth base.
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