Daniel l. Rubinfeld
Daniel l. Rubinfeld
Daniel l. Rubinfeld
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152 Part 2 Producers, Consumers, and Competitive Markets<br />
5 Choice Under Uncertainty<br />
53<br />
Probability<br />
OUTCOME 1<br />
DEVIATION<br />
SQUAREO<br />
OUTCOME 2<br />
DEViATION<br />
SQUAREO<br />
AVERAGE<br />
DEVIATION<br />
SQUARED<br />
STANDARD<br />
DEVIATION<br />
0.2<br />
Job 1 2000<br />
Job 2 1510<br />
250,000<br />
100<br />
1000<br />
510<br />
250,000<br />
980,100<br />
250,000<br />
9,900<br />
500<br />
99.50<br />
01<br />
Job 2<br />
Job 1<br />
standard deviation Square<br />
root of the a\'erage of the<br />
squares of the de\'iations of<br />
the payoffs associated with<br />
each outcome fro111 their<br />
expected values.<br />
By themseh'es, de\'iations do not provide a measure of \'ariability Why<br />
Because they are sometimes positive and sometimes negative, and as you can<br />
see from Table 5.2, the a\'erage de\'iation is always 0. 2 To get around this problem,<br />
\ve square each de\·iation, yielding numbers that are always positive. We<br />
then measure variability by calculating the standard deviation: the square root<br />
of the average of the sqlLnrcs of the deviations of the payoffs associated with each<br />
outcome from their expected value 3<br />
Table 5.3 shows the calculation of the standard deviation for our example.<br />
Note that the average of the squared de\'iations under Job 1 is given by<br />
5(5250,000) + 5($250,000) = $250,000<br />
The standard deviation is therefore equal to the square root of 5250,000, or 5500.<br />
Likewise, the a\'crage of the squared de\'iations under Job 2 is given by<br />
.99($100) + .01(5980,100) = 59900<br />
The standard deviation is the square root of 59,900, or 599.50. Thus the second<br />
job is much less risky than the first; the standard deviation of the incomes is<br />
much lo'wer.~<br />
The concept of standard deviation applies equally \vell when there are many<br />
outcomes rather than just two. Suppose, for example, that the first job yields<br />
incomes ranging from 51000 to 52000 in increments of $100 that are all equally<br />
likely. The second job yields incomes from $1300 to $1700 (again in increments of<br />
5100) that are also equally likely. Figure 5.1 sho'ws the alternatives graphically. (If<br />
there had been only two equally probable outcomes, then the figure would be<br />
drawn as two vertical lines, each 'with a height of 0.5.)<br />
You can see from Figure 5.1 that the first job is riskier than the second. The<br />
"spread" of possible payoffs for the first job is much greater than the spread for<br />
the second. As a result, the standard deviation of the payoffs associated with the<br />
first job is greater than that associated \vith the second.<br />
In this particular example, all payoffs are equally likely. Thus the curves<br />
describing the probabilities for each job are flat. In rnany cases, hO\·..,e\'er, some<br />
51000 51300 52000 Income<br />
The dish-ibution of payoffs associated with Job 1 has a o-reater spread and a £reater<br />
standard deviation than the distribution of payoffs associated with Job 2. Botl; distributions<br />
are flat because all outcomes are equally likely.<br />
- &¥i&& ~ ±<br />
payoffs are more likely than others. Figme 5.2 shows a situation in which the<br />
most extrerne payoffs are the least likely. Again, the salary from Job 1 has a<br />
greater standard deviation. From this point on, we will use the standard deviation<br />
of payoffs to measure degree of risk.<br />
Decision Mal