Daniel l. Rubinfeld

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=77 2 Producers, Consumers, and Competitive Markets To describe risk quantitati\'ely, we begin by listing all the possible outcomes of a particular action or e\'ent as well as the likelihood that each outcome will occur.l Suppose, for example, that you are considering investing in a company that explores for offshore oil. If the exploration effort is successful, the company's stock will increase from 530 to 540 per share; if not, the price ,yill fall to S20 per share. Thus there are two possible fuhlre outcomes: a S40-per-share price and a S20-per-share price. Our offshore oil exploration example had two possible outcomes: Success yields a payo~f of $40 per share, failure a payoff o~ $20. per share. Denoting ~lprobability ot" by Pr, ,Ye express the expected value 111 thIS case as Expected value = Pr(success)($40/share) + Pr(failure)($20/share) = (1/4)($40/share) + (3/4)(S20/share) = S25/share More generally, if there are two possible outcomes having payoffs Xl and X 2 and if the probabilities of each outcome are given by Prj and Pr2, then the expected value is 5 Choice Under Uncertainty 5 probability Likelihood that a given outcome will OCCUL expected value Probabilityweighted awrage of the values associated with all possible outcomes. payoff Value associated with a possible outcome. Probability is the likelihood that a given outcome will occur. In our example, the probability that the oil exploration project is successful might be 1/4 and the probability that it is unsuccessful 3/4. (Note that the probabilities for all possible events must sum to L) Our interpretation of probability can depend on the nature of the uncertain event, on the beliefs of the people involved, or both. One objective interpretation of probability relies on the frequency with which certain events tend to occur. Suppose we kno\\' that of the last 100 offshore oil explorations, 25 have succeeded and 75 failed. In that case, the probability of success of 1/4 is objective because it is based directly on the frequency of similar experiences. But what if there are no similar past experiences to help measure probability In such instances, objective measures of probability cannot be deduced and more subjective measures are needed. 5ubjectiue probllbility is the perception that an outcome will occur. This perception may be based on a person's judgment or experience, but not necessarily on the frequency with which a particular outcome has actually occurred in the past. When probabilities are subjectively determined, different people may attach different probabilities to different outcomes and thereby make different choices. For example, if the search for oil were to take place in an area where no previous searches had ever occurred, I might attach a higher subjective probability than you to the chance that the project will succeed: Perhaps I knovv more about the project or I have a better understanding of the oil business and can therefore make better use of our common information. Either different information or different abilities to process the same information can cause subjective probabilities to vary among individuals. Regardless of the interpretation of probability, it is used in calculating two important measures that help us describe and compare risky choices. One measure tells us the expected villue and the other the vnrillbility of the possible outcomes. Expected Value The expected value associated with an uncertain situation is a 'weighted a\'erage of the payoffs or values resulting from all possible outcomes. The probabilities of each outcome are used as weights. Thus the expected value measures the central telldency-that is, the payoff or value that I·ve would expect on average. VVhen there are 11 possible outcomes, the expected value becomes Variability Variability is the extent to which the possible outcomes of an uncertain situation differ. To see why variability is important, suppose you are choosing betvveen two part-time sales jobs that have the same expected income ($1500). The first job is based entirely on commission-the income earned depends on how much you sell. There are two equally likely payoffs for this job: $2000 for a successful sales effort and $1000 for one that is less successful. The second job is salaried. It is very likely (.99 probability) that you will earn $1510, but there is a .01 probability that the company will go out of business, in which case you would earn $510 in severance pay. Table 5.1 summarizes these possible outcomes, their payoffs, and their probabilities. Note that these two jobs have the same expected income. For Job 1, expected income is .5(52000) + .5($1000) = $1500; for Job 2 it is .99($1510) + .01($510) = $1500. Hovvever, the vilrillbility of the possible payoffs is different. We measure variability by recognizing that large differences behveen actual and expected payoffs (whether positive or negative) irnply greater risk. We call these differences deviations. Table 5.2 shows the deviations of the possible incomes from the expected income from each of the hvo jobs. OUTCOME 1 OUTCOME 2 Expected Probability Income ($) Probability Income ($) Income ($) Job 1: Commission .5 2000 .5 1000 1500 Job 2: Fixed salary .99 1510 .01 510 1500 variability Extent to which possible outcomes of an uncertain event may differ. deviation Difference between expected payoff and actual payoff. I Some people distinguish between uncertainty and risk along the lines suggested some 60 years ago by economist Frank Knight Uncertainty can refer to situations in which many outcomes are R Os - sible but their likelihoods unknown. Risk then refers to situations in which we can list all possIble outcomes and know the likelihood of each occurring. In this chapter, we will always refer to risky situations but \\"ill simplify the discussion by using lIncertainty and risk interchangeably OUTCOME 1 DEVIATION OUTCOME 2 DEVIATION Job 1 2000 500 1000 -500 Job 2 1510 10 510 -990
152 Part 2 Producers, Consumers, and Competitive Markets 5 Choice Under Uncertainty 53 Probability OUTCOME 1 DEVIATION SQUAREO OUTCOME 2 DEViATION SQUAREO AVERAGE DEVIATION SQUARED STANDARD DEVIATION 0.2 Job 1 2000 Job 2 1510 250,000 100 1000 510 250,000 980,100 250,000 9,900 500 99.50 01 Job 2 Job 1 standard deviation Square root of the a\'erage of the squares of the de\'iations of the payoffs associated with each outcome fro111 their expected values. By themseh'es, de\'iations do not provide a measure of \'ariability Why Because they are sometimes positive and sometimes negative, and as you can see from Table 5.2, the a\'erage de\'iation is always 0. 2 To get around this problem, \ve square each de\·iation, yielding numbers that are always positive. We then measure variability by calculating the standard deviation: the square root of the average of the sqlLnrcs of the deviations of the payoffs associated with each outcome from their expected value 3 Table 5.3 shows the calculation of the standard deviation for our example. Note that the average of the squared de\'iations under Job 1 is given by 5(5250,000) + 5($250,000) = $250,000 The standard deviation is therefore equal to the square root of 5250,000, or 5500. Likewise, the a\'crage of the squared de\'iations under Job 2 is given by .99($100) + .01(5980,100) = 59900 The standard deviation is the square root of 59,900, or 599.50. Thus the second job is much less risky than the first; the standard deviation of the incomes is much lo'wer.~ The concept of standard deviation applies equally \vell when there are many outcomes rather than just two. Suppose, for example, that the first job yields incomes ranging from 51000 to 52000 in increments of $100 that are all equally likely. The second job yields incomes from $1300 to $1700 (again in increments of 5100) that are also equally likely. Figure 5.1 sho'ws the alternatives graphically. (If there had been only two equally probable outcomes, then the figure would be drawn as two vertical lines, each 'with a height of 0.5.) You can see from Figure 5.1 that the first job is riskier than the second. The "spread" of possible payoffs for the first job is much greater than the spread for the second. As a result, the standard deviation of the payoffs associated with the first job is greater than that associated \vith the second. In this particular example, all payoffs are equally likely. Thus the curves describing the probabilities for each job are flat. In rnany cases, hO\·..,e\'er, some 51000 51300 52000 Income The dish-ibution of payoffs associated with Job 1 has a o-reater spread and a £reater standard deviation than the distribution of payoffs associated with Job 2. Botl; distributions are flat because all outcomes are equally likely. - &¥i&& ~ ± payoffs are more likely than others. Figme 5.2 shows a situation in which the most extrerne payoffs are the least likely. Again, the salary from Job 1 has a greater standard deviation. From this point on, we will use the standard deviation of payoffs to measure degree of risk. Decision Mal
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Daniel l. Rubinfeld

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