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Chapter 6
Choice Under Uncertainty
Up until now, we have been concerned with choice under certainty. A consumer chooses which commodity bundle to consume. A producer chooses how much output to produce using which mix of inputs. In either case, there is no uncertainty about the outcome of the choice. We now turn to considering choice under uncertainty, where the objects of choice are not certainties, but distributions over outcomes. For example, suppose that you have a choice between two alternatives. Under alternative A, you roll a six-sided die. If the die comes up 1, 2, or 3, you get $1000. If it comes up 4, 5, or 6, you lose $300. Under alternative B, you choose a card from a standard 52 card deck. If the card you choose is black, you pay me $200. If it is a heart, you get a free trip to Bermuda. If it is a diamond, you have to shovel the snow off of my driveway all winter. If I were to ask you whether you preferred alternative A or alternative B, you could probably tell me. Indeed, if I were to write down any two random situations, call them L 1 and L 2 , you could probably tell me which one you prefer. And, there is even the possibility that your preferences would be complete, transitive (i.e., rational), and continuous. If this is true then I can come up with a utility function representing your preferences over random situations, call it U (L), such that L 1 is strictly preferred to L 2 if and only if U (L 1 ) > U (L 2 ). Thus, without too much effort, we can extend our standard utility theory to utility under uncertainty. All we need is for the consumer to have well defined preferences over uncertain alternatives. Now, recall that I said that much of what we do from a modeling perspective is add structure to people’s preferences in order to be able to say more about how they behave. In this situation, what we would like to be able to do is say that a person’s preferences over uncertain alternatives
should be able to be expressed in terms of the utility the person would assign to the outcome if it were certain to occur, and the probability of that outcome occurring. For example, suppose we are considering two different uncertain alternatives, each of which offers a different distribution over three outcomes: I buy you a trip to Bermuda, you pay me $500, or you paint my house. The probability of each outcome under alternatives A and B are given in the following table:
Bermuda -$500 Paint my house A .3 .4. B .2 .7. What we would like to be able to do is express your utility for these two alternatives in terms of the utility you assign to each individual outcome and the probability that they occur. For example, suppose you assign value uB to the trip to Bermuda, um to paying me the money, and up to painting my house. It would be very nice if we could express your utility for each alternative by multiplying each of these numbers by the probability of the outcome occurring, and summing. That is:
U (A) = 0. 3 uB + 0. 4 um + 0. 3 up U (B) = 0. 2 uB + 0. 7 um + 0. 1 up.
Note that if this were the case, we could express the utility of any distribution over these outcomes in the same way. If the probabilities of Bermuda, paying me the money, and painting my house are pB , pm, and pp, respectively, then the expected utility of the alternative is
pB uB + pmum + ppup.
This would be very useful, since it would allow us to base our study of choice under uncertainty on a study of choice over certain outcomes, extended in a simple way. However, while the preceding equation, known as an expected utility form, is useful, it is not necessarily the case that a consumer with rational preferences over uncertain alternatives will be such that those alternatives can be represented in this form. Thus the question we turn to first is what additional structure we have to place on preferences in order to ensure that a person’s preferences can be represented by a utility function that takes the expected utility form. After identifying these conditions, we will go on to show how utility functions of the expected utility form can be used to study behavior under uncertainty, and draw testable implications about people’s behavior that are not implied by the standard approach.
If L and L^0 are lotteries, a compound lottery over these two lotteries can be represented as aL + (1 − a) L^0 , where 0 ≤ a ≤ 1 is the probability of lottery L occurring.
6.1.1 Preferences Over Lotteries
We begin by building up a theory of rational preferences over lotteries. Once we do that, we’ll know that there is a utility function that represents those preferences (under certain conditions). We’ll then go on to ask whether those preferences can be represented by a utility function of the expected utility form. Let L˜ be the set of all possible lotteries. Thus L˜ is like X from consumer theory, the set of all possible consumption bundles. We want our consumer to have rational preferences over lotteries. So, suppose that the relation % represents strict preferences over lotteries, and suppose that these preferences are rational, i.e., complete and transitive. We will also assume that the consumer’s preferences are consequentialist. Basically, this means that consumers care only about the distribution over final outcomes, not whether this dis- tribution comes about as a result of a simple lottery, or a compound lottery. In other words, the consumer is indifferent between any two compound lotteries that can be reduced to the same simple lottery. This property is often called reduction of compound lotteries.^1 Because of the reduction property, we can confine our attention to the set of all simple lotteries, and from now on we will let L˜ be the set of all simple lotteries. The other requirement we needed for preferences to be representable by a utility function was continuity. In consumer theory, continuity meant that you could not pass from the lower level set of a consumer’s utility function to the upper level set without passing through the indifference set. Something similar is involved with continuity here, but what we are interested in is continuity in probabilities. Suppose that L Â L^0. The essence of continuity is that adding a sufficiently small probability of some other lottery, L^00 , to L should not reverse this preferences. That is:
if L Â L^0 , then L Â (1 − a) L^0 + aL^00 for some a > 0. (^1) Another way to think about the reduction property is that we’re assuming there is no process-oriented utility. Consumers do not enjoy the process of the gamble, only the outcome, eliminating the “fun of the gamble” in settings like casinos.
Formally, % on L˜ is continuous if for any L, L^0 , and L^00 , and any a ∈ (0, 1), the sets:
©a ∈ (0, 1) |L % (1 − a) L (^0) + aL 00 ª
and ©a ∈ (0, 1) | (1 − a) L (^0) + aL (^00) % Lª
are closed. Continuity is mostly a technical assumption needed to get the existence of a utility function. But, notice that its validity is less compelling than it was in the certainty case. For example, suppose L is a trip to Bermuda and L^0 is $3000, and that you prefer L^0 to L. Now suppose we introduce L^00 , which is violent, painful, death. If preferences are continuous, then the trip to Bermuda should also be less preferred than $3000 with probability 1 − a and violent painful death with probability a, provided that a is sufficiently small. For many people, there is no probability a > 0 such that this would be the case, even though when a = 0, L^0 is preferred to L. If the consumer has rational, continuous preferences over L,˜ we know that there is a utility function U () such that U () represents those preferences. In order to get a utility function of the expected utility form that represents those preferences, the consumer’s preferences must also satisfy the independence axiom. The preferences relation % on L˜ satisfies the independence axiom if for all L, L^0 , and L^00 and a ∈ (0, 1), L % L^0 if and only if aL + (1 − a) L^00 % aL^0 + (1 − a) L^00. The essence of the independence axiom is that outcomes that occur with the same probability under two alternatives should not affect your preferences over those alternatives. For example, suppose that I offer you the choice between the following two alternatives:
L : $5 with probability 15 , 0 with probability (^45) L^0 : $12 with probability 101 , 0 with probability 109.
Suppose you prefer L to L^0. Now consider the following alternative. I flip a coin. If it comes up heads, I offer you the choice between L and L^0. If it comes up tails, you get nothing. What the independence axiom says is that if I ask you to choose either L or L^0 before I flip the coin, your preference should be the same as it was when I didn’t flip the coin. That is, if you prefer L to L^0 , you should also prefer 12 L + 12 0 % 12 L^0 + 12 0 , where 0 is the lottery that gives you 0 with probability
Thus if this person satisfies the independence axiom, L 2 should be preferred to L^02 whenever LA is preferred to LB , which is the same as in the L 1 vs. L^01 case above. Hence if L 1 is preferred to L^01 , then L 2 should also be preferred to L^02. Usually, about half of the people prefer L 1 to L^01 but L^02 to L 2. Does this mean that they are irrational? Not really. What it means is that they do not satisfy the independence axiom. Whether or not such preferences are irrational has been a subject of debate in economics. Some people think yes. Others think no. Some people would argue that if your preferences don’t satisfy the independence axiom, it’s only because you don’t understand the problem. And, once the nature of your failure has been explained to you, you will agree that your behavior should satisfy the independence axiom and that you must have been mistaken or crazy when it didn’t. Others think this is complete nonsense. Basically, the independence axiom is a source of great controversy in economics. This is especially true because the independence axiom leads to a great number of paradoxes like the Allais paradox mentioned earlier. In the end, the usefulness of the expected utility framework that we are about to develop usually justifies its use, even though it is not perfect. A lot of the research that is currently going on is trying to determine how you can have an expected utility theory without the independence axiom.
6.1.2 The Expected Utility Theorem
We now return to the question of when there is a utility function of the expected utility form that represents the consumer’s preferences. Recall the definition:
Definition 9 A utility function U (L) has an expected utility form if there are real numbers u 1 , ..., uN such that for every simple lottery L = (p 1 , ..., pN ),
U (L) =
X
n
pnun.
The reduction property and the independence axiom combine to show that utility function U (L) has the expected utility form if and only if it is linear, meaning it satisfies the property:
U
à XK
k=
tkLk
X^ K
k=
tkU (Lk) (6.1)
for any K lotteries. To see why, note that we need to show this in “two directions” - first, that the expected utility form implies linearity; then, that linearity implies the expected utility form.
- Suppose that U (L) has the expected utility form. Consider the compound lottery PKk=1 tkLk.
U
à XK
k=
tkLk
X
n
un
à XK
k=
tkpkn
X^ K
k=
tk
ÃX
n
unpkn
X^ K
k=
tkU (Lk). So, it is linear.
- Suppose that U (L) is linear. Let Ln^ be the lottery that awards outcome an with probability
- Then U (L) = U
ÃX
n
pnLn
X
n
pnU (Ln) =
X
n
pnun. So, it has the expected utility form.
Thus proving that a utility function has the expected utility form is equivalent to proving it is linear. We will use this fact momentarily. The expected utility theorem says that if a consumer’s preferences over simple lotteries are rational, continuous, and exhibit the reduction and independence properties, then there is a utility function of the expected utility form that represents those preferences. The argument is by construction. To make things simple, suppose that there is a best prize, aB , and a worst prize, aW , among the prizes. Let LB^ be the “degenerate lottery” that puts probability 1 on aB occurring, and LW^ be the degenerate lottery that puts probability 1 on aW. Now, consider some lottery, L, such that LB^ Â L Â LW^. By continuity, there exists some number, aL, such that
aLLB^ + (1 − aL) LW^ ∼ L.
We will define the consumer’s utility function as U (L) = aL as defined above, and note that U ¡LB^ ¢^ = 1 and U ¡LW^ ¢^ = 0. Thus the utility assigned to a lottery is equal to the probability put on the best prize in a lottery between the best and worst prizes such that the consumer is indifferent between L and aLLB^ + (1 − aL) LW^. In order to show that U (L) takes the expected utility form, we must show that: U ¡tL + (1 − t) L^0 ¢^ = taL + (1 − t) aL 0.
If this is so, then U () is linear, and thus we know that it can be represented in the expected utility form. Now, L ∼ aLLB^ + (1 − aL) LW^ , and L^0 ∼ aL 0 LB^ + (1 − aL 0 ) LW^. Thus:
U ¡tL + (1 − t) L^0 ¢ = U ¡t ¡aLLB^ + (1 − aL) LW^ ¢^ + (1 − t) ¡aL 0 LB^ + (1 − aL 0 ) LW^ ¢¢
The Expected Utility Form is preserved only by positive linear transformations. If U () and V () are utility functions representing %, and U () has the expected utility form, then V () also has the expected utility form if and only if there are numbers a > 0 and b such that:
U (L) = aV (L) + b.
In other words, the expected utility property is preserved by positive linear (affine) transformations, but any other transformation of U () does not preserve this property. MWG calls the utility function of the expected utility form a von-Neumann-Morgenstern (vNM) utility function, and I’ll adopt this as well. That said, it is important that we do not confuse the vNM utility function, U (L) , with the numbers u 1 , ..., uN associated with it.^3 An important consequence of the fact that the expected utility form is preserved only by positive linear transformations is that a vNM utility function imposes cardinal significance on utility. To see why, consider the utility associated with four prizes, u 1 , u 2 , u 3 , and u 4 , and suppose that
u 1 − u 2 > u 3 − u 4.
Suppose we apply a positive linear transformation to these numbers:
vn = aun + b.
Then
v 1 − v 2 = au 1 + b − (au 2 + b) = a (u 1 − u 2 )
a (u 3 − u 4 ) = au 3 + b − (au 4 + b) = v 3 − v 4.
Thus v 1 − v 2 > v 3 − v 4 if and only if u 1 − u 2 > u 3 − u 4. And, since any utility function of the expected utility form that represents the same preferences will exhibit this property, differences in utility numbers are meaningful. The numbers assigned by the vNM utility function have cardinal significance. This will become important when we turn to our study of utility for money and risk aversion, which we do next.
6.1.3 Constructing a vNM utility function
Let A = {a 1 , ..., aN } denote the set of prizes. Suppose that aN is the best prize and a 1 is the worst prize. We are going to show how to construct numbers u 1 , ..., uN that make up a utility function with the expected utility form that represents preferences over these prizes. (^3) Later, when we allow for a continuum of prizes (such as monetary outcomes), the numbers u 1 , ..., uN become the function u (x), and we’ll call the lowercase u (x) function the Bernoulli utility function.
First, we are going to arbitrarily choose uN = 1 and u 1 = 0. Why? Because we can. Remember, the point here is to construct a utility function that has the expected utility form. We could just as easily do it for arbitrary specifications of uN and u 1 , but this is notationally a bit simpler. Now, for each prize ai, define ui to be the probability such that the decision maker is indifferent between prize ai for sure and a lottery that offers aN with probability ui and a 1 with probability 1 −ui. Let’s refer to the lottery that offers prize aN with probability ui and prize a 1 with probability (1 − ui) lottery Si. So, ui’s are between 0 and 1. Notice that if we specify the numbers in this way it makes sense that uN = 1 and u 1 = 0, since the decision maker should be indifferent between the best prize for sure and a lottery that offers the best prize with probability u 1 = 1, etc. This gives us a way of defining numbers ui. Now, we want to argue that this way of defining the ui’s, combined with consequentialism (reduction of compound lotteries) and Independence of Irrelevant alternatives yields a utility function that looks like U (L) = P^ piui. So, consider lottery L = (p 1 , ..., pN ). This lottery offers prize ai with probability pi. But, we know that the decision maker is indifferent between ai for sure and a lottery that offers prize^ aN with probability ui and price a 1 with probability 1 −ui. Thus, using IIA, we know that the decision maker is indifferent between lottery L and a compound lotter in which, with probability pi, the decision maker faces another lottery: uN with probability ui and u 1 with probability 1 − ui. This lottery is depicted as L^0 in the following diagram. Note that L^0 only has two distinct prizes: aN and a 1. By reduction of compound lotteries, we can combine the total probability of each outcome, making an equivalent simple lottery, L^00. The utility for lottery L^00 is (P^ piui) uN +(1 − P^ piui) u 1. Since uN = 1 and u 1 = 0, this gives that U (L) = U (L^0 ) = U (L^00 ) = P^ piui, which is what we wanted to show. Defining utility in this way gives us a representation with the expected utility form.
6.2 Utility for Money and Risk Aversion
The theory of choice under uncertainty is most frequently applied to lotteries over monetary out- comes. The easiest way to treat monetary outcomes here is to let x be a continuous variable representing the amount of money received. With a finite number of outcomes, assign a number un to each of the N outcomes. We could also do this with the continuous variable, x, just by letting ux be the number assigned to the lottery that assigns utility x with probability 1. In this
case, there would be one value of ux for each real number, x. But, this is just what it means to be a function. So, we’ll let the function u (x) play the role that un did in the finite outcome case. Thus u (x) represents the utility associated with the lottery that awards the consumer x dollars for sure. Since there is a continuum of outcomes, we need to use a more general probability structure as well. With a discrete number of outcomes, we represented a lottery in terms of a vector (p 1 , ..., pN ), where pn represents the probability of outcome n. When there is a continuum of outcomes, we will represent a lottery as a distribution over the outcomes. One concept that you are probably familiar with is using a probability density function f (x). When we had a finite number of outcomes, we denoted the probability of any particular outcome by pn. The analogue to this when there are a continuous number of outcomes is to use a probability density function (pdf). The pdf is defined such that: Pr (a ≤ x ≤ b) =
Z (^) b a f (x) dx.
Recall that when a distribution can be represented by a pdf, it has no atoms (discrete values of x with strictly positive probability of occurring). Thus the probability of any particular value of x being drawn is zero. The expected utility of a distribution f () is given by:
U (f ) =
Z +∞
−∞^ u^ (x)^ f^ (x)^ dx, which is just the continuous version of U (L) = P n pnun. In order to keep things straight, we will call u (x) the Bernoulli utility function, while we will continue to refer to U (f ) as the vNM utility function. It will also be convenient to write a lottery in terms of its cumulative distribution function (cdf) rather than its pdf. The cdf of a random variable is given by:
F (b) =
Z (^) b −∞ f (x) dx.
When we use the cdf to represent the lottery, we’ll write the expected utility of F as: Z (^) +∞ −∞ u (x) dF (x).
Mathematically, the latter formulation lets us deal with cases where the distribution has atoms, but we aren’t going to worry too much about the distinction between the two. The Bernoulli utility function provides a convenient way to think about a decision maker’s attitude toward risk. For example, consider a gamble that offers $100 with probability 12 and 0
with probability 12. Now, if I were to offer you the choice between this lottery and c dollars for sure, how small would c have to be before you are willing to accept the gamble? The expected value of the gamble is 12 100 + 12 0 = 50. However, if offered the choice between 50 for sure and the lottery above, most people would choose the sure thing. It is not until c is somewhat lower than 50 that many people find themselves willing to accept the lottery. For me, I think the smallest c for which I am willing to accept the gamble is 40. The fact that 40 < 50 captures the idea that I am risk averse. My expected utility from the lottery is less than the utility I would receive from getting the expected value of the gamble for sure. The minimum amount c such that I would accept the gamble instead of the sure thing is known as the certainty equivalent of the gamble, since it equals the certain amount of money that offers the same utility as the lottery. The difference between the expected value of the lottery, 50, and my certainty equivalent, 40, is known as my risk premium, since I would in effect be willing to pay somebody 10 to take the risk away from me (i.e. replace the gamble with its expected value). Formally, let’s define the certainty equivalent. Let c (F, u) be the certainty equivalent for a person with Bernoulli utility function u facing lottery F , defined according to:
u (c (F, u)) =
Z
u (x) dF (x) Although generally speaking people are risk averse, this is a behavioral postulate rather than an assumption or implication of our model. But, people need not be risk averse. In fact, we can divide utility functions into four classes:
- Risk averse. A decision maker is risk averse if the expected utility of any lottery, F , is not more than the utility of the getting the expected value of the lottery for sure. That is, if: Z u (x) dF (x) ≤ u
μZ xdF (x)
for all F. (a) If the previous inequality is strict, we call the decision maker strictly risk averse. (b) Note also that since u (c (F, u)) = R^ u (x) dF (x) and u () is strictly increasing, an equiv- alent definition of risk aversion is that the certainty equivalent c (F, u) is no larger than the expected value of the lottery, R^ xdF (x) for any lottery F.
- Risk loving. A decision maker is risk loving if the expected utility of any lottery is not less than the utility of getting the expected value of the lottery for sure: Z u (x) dF (x) ≥ u
μZ xdF (x)
the manifestation of risk aversion in panel a is in the fact that the dotted line between (1, u (1)) and (3, u (3)) lies everywhere below the utility function. To see if you understand, draw the diagram for a risk-loving decision maker, and convince yourself that 12 u (1) + 12 u (3) > u ¡^12 1 + 12 3 ¢.
6.2.1 Measuring Risk Aversion: Coefficients of Absolute and Relative Risk
Aversion
As we said, risk aversion is equivalent to concavity of the utility function. Thus one would expect that one utility function is “more risk averse” than another if it is “more concave.” While this is true, it turns out that measuring the risk aversion is more complicated than you might think (isn’t everything in this course?). Actually, it is only slightly more complicated. You might be tempted to think that a good measure of risk aversion is that Bernoulli utility function u 1 () is more risk averse than Bernoulli utility function u 2 () if |u^001 ()| > |u^002 ()| for all x. However, there is a problem with this measure, in that it is not invariant to positive linear transformations of the utility function. To see why, consider utility function u 1 (x), and apply the linear transformation u 2 () = au 1 () + b, where a > 1. We know that such a transformation leaves the consumer’s attitudes toward risk unchanged. However, u^002 () = au^001 () > u^001 (). Thus if we use the second derivative of the Bernoulli utility function as our measure of risk aversion, we find that it is possible for a utility function to be more risk averse than another, even though it represents the exact same preferences. Clearly, then, this is not a good measure of risk aversion. The way around the problem identified in the previous paragraph is to normalize the second derivative of the utility function by the first derivative. Using u 2 () from the previous paragraph, we then get that: u^002 () u^02 () =^
au^001 () au^01 () =^
u^001 () u^01 (). Thus this measure of risk aversion is invariant to linear transformations of the utility function. And, it’s almost the measure we will use. Because u^00 < 0 for a concave function, we’ll multiply by − 1 so that the risk aversion number is non-negative for a risk-averse consumer. This gives us the following definition:
Definition 10 Given a twice-differentiable Bernoulli utility function u (), the Arrow-Pratt measure of absolute risk aversion is given by:
rA (x) = − u
(^00) (x) u^0 (x).
Note the following about the Arrow-Pratt (AP) measure:
- rA (x) is positive for a risk-averse decision maker, 0 for a risk-neutral decision maker, and negative for a risk-loving decision maker.
- rA (x) is a function of x, where x can be thought of as the consumer’s current level of wealth. Thus we can admit the situation where the consumer is risk averse, risk loving, or risk neutral for different levels of initial wealth.
- We can also think about how the decision maker’s risk aversion changes with her wealth. How do you think this should go? Do you become more or less likely to accept a gamble that offers 100 with probability 12 and − 50 with probability 12 as your wealth increases? Hopefully, you answered more. This means that you become less risk averse as wealth increases, and this is how we usually think of people, as having non-increasing absolute risk aversion.
- The AP measure is called a measure of absolute risk aversion because it says how you feel about lotteries that are defined over absolute numbers of dollars. A gamble that offers to increase or decrease your wealth by a certain percentage is a relative lottery, since its prizes are defined relative to your current level of wealth. We also have a measure of relative risk aversion, rR (x) = − xu
(^00) (x) u^0 (x). But, we’ll come back to that later.
6.2.2 Comparing Risk Aversion
Frequently it is useful to know when one utility function is more risk averse than another. For example, risk aversion is important in the study of insurance, and a natural question to ask is how a person’s desire for insurance changes as he becomes more risk averse. Fortunately, we already have the machinery in place for our comparisons. We say that utility function u 2 () is at least as risk averse as u 1 () if any of the following hold (in fact, they are all equivalent):
- c (F, u 2 ) ≤ c (F, u 1 ) for all F.
- rA (x, u 2 ) ≥ rA (x, u 1 )
- u 2 () can be derived from u 1 () by applying an increasing, concave transformation, i.e., u 2 () = g (u 1 (x)), where g () is increasing and concave. Note, this is what I meant when I said being
mentioned earlier, the natural assumption to make (since it corresponds with how people actually seem to behave) is that people becomes less risk averse as they become wealthier. In terms of the measures we have for risk aversion, we say that a person exhibits non-increasing absolute risk aversion whenever rA (x) is non-increasing in x. In MWG Proposition 6.C.3, there are some alternate definitions. Figuring them out would be a useful exercise. Of particular interest, I think, is part iii), which says that having non-increasing (or decreasing) absolute risk aversion means that as your wealth increases, the amount you are willing to pay to get rid of a risk decreases. What does this say about insurance? Basically, it means that the wealthy will be willing to pay less for insurance and will receive less benefit from being insured. Formalizing this, let z be a random variable with distribution F and a mean of 0. Thus z is the prize of a lottery with distribution F. Let cx (the certainty equivalent) be defined as:
u (cx) =
Z
u (x + z) dF (z).
If the utility function exhibits decreasing absolute risk aversion, then x − cx (corresponding to the premium the person is willing to pay to get rid of the uncertainty) will be decreasing in x. As before, it is natural to think of people as exhibiting nonincreasing relative risk aversion. That is, they are more likely to accept a proportional gamble as their initial wealth increases. Although the concept of relative risk aversion is useful in a variety of contexts, we will primarily be concerned with absolute risk aversion. One reason for this that many of the techniques we develop for studying absolute risk aversion translate readily to the case of relative risk aversion.
6.2.3 A Note on Comparing Distributions: Stochastic Dominance
We aren’t going to spend a lot of time talking about comparing different distributions in terms of their risk and return because these are concepts that involve slightly more knowledge of probability and will most likely be developed in the course of any applications you see that use them. However, I will briefly mention them. Suppose we are interested in knowing whether one distribution offers higher returns than an- other. There is some ambiguity as to what this means. Does it mean higher average monetary return (i.e., the mean of F ), or does it mean higher expected utility? In fact, when a consumer is risk averse, a distribution with a higher mean may offer a lower expected utility if it is riskier. For example, a sufficiently risk averse consumer will prefer x = 1. 9 for sure to a 50-50 lottery over 1 and 3. This is true even though the mean of the lottery, 2, is higher than the mean of the sure
thing, 1.9. Thus if we are concerned with figuring out which of two lotteries offers higher utility than another, simply comparing the means is not enough. It turns out that the right concept to use when comparing the expected utility of two distribu- tions is called first-order stochastic dominance (FOSD). Consider two distribution functions, F () and G (). We say that F () first-order stochastically dominates G () if F (x) ≤ G (x) for all x. That is, F () FOSD G () if the graph of F () lies everywhere below the graph of G (). What is the meaning of this? Recall that F (y) gives the probability that the lottery offers a prize that is less than or equal to y. Thus if F (x) ≤ G (x) for all x, this means that for any prize, y, the probability that G ()’s prize is less than or equal to y is greater than the probability that F ()’s prize is less than or equal to y. And, if it is the case that F () FOSD G (), it can be shown that any consumer with a strictly increasing utility function u () will prefer F () to G (). That is, as long as you prefer more money to less, you will prefer lottery F () to G (). Now, it’s important to point out that most of the time you will not be able to rank distributions in terms of FOSD. It will need not be the case that either F () FOSD G () or G () FOSD F (). In particular, the example from two paragraphs ago ( 1. 9 for sure vs. 1 or 3 with equal probability) cannot be ranked. As in the case of ranking the risk aversion of two utility functions, the primary use of this concept is in figuring out (in theory) how a decision maker would react when the distribution of prizes “gets higher.” FOSD is what we use to capture the idea of “gets higher.” And, knowing an initial distribution F () , FOSD gives us a good guide to what it means for the distribution to get higher: The new distribution function must lay everywhere below the old one. So, FOSD helps us formalize the idea of a distribution “getting higher.” In many circumstances it is also useful to have a concept of “getting riskier.” The concept we use for “getting riskier” is called second-order stochastic dominance (SOSD). One way to understand SOSD is in terms of mean preserving spreads. Let X be a random variable with distribution function F (). Now, for each value of x, add a new random variable zx, where zx has mean zero. Thus zx can be thought of as a noise term, where the distribution of the noise depends on x but always has a mean of zero. Now consider the random variable y = x + zx. Y will have the same mean as X, but it will be riskier because of all of the noise terms we have added in. And, we say that for any Y than has the same mean as X and can be derived from X by adding noise, Y is riskier than X. Thus, we say that X second-order stochastically dominates Y. Let me make two comments at this point. First, as usual, it won’t be the case that for any two random variables (or distributions), one must SOSD the other. In most cases, neither will SOSD.
