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CHAPTER 12
UNCERTAINTY
Uncertainty is a fact of life. People face risks every time they take a shower, walk across the street, or make an investment. But there are financial insti- tutions such as insurance markets and the stock market that can mitigate at least some of these risks. We will study the functioning of these mar- kets in the next chapter, but first we must study individual behavior with respect to choices involving uncertainty.
12.1 Contingent Consumption
Since we now know all about the standard theory of consumer choice, let’s try to use what we know to understand choice under uncertainty. The first question to ask is what is the basic “thing” that is being chosen? The consumer is presumably concerned with the probability distri- bution of getting different consumption bundles of goods. A probability distribution consists of a list of different outcomes—in this case, consump- tion bundles—and the probability associated with each outcome. When a consumer decides how much automobile insurance to buy or how much to
218 UNCERTAINTY (Ch. 12)
invest in the stock market, he is in effect deciding on a pattern of probability distribution across different amounts of consumption. For example, suppose that you have $100 now and that you are con- templating buying lottery ticket number 13. If number 13 is drawn in the lottery, the holder will be paid $200. This ticket costs, say, $5. The two outcomes that are of interest are the event that the ticket is drawn and the event that it isn’t. Your original endowment of wealth—the amount that you would have if you did not purchase the lottery ticket—is $100 if 13 is drawn, and $ if it isn’t drawn. But if you buy the lottery ticket for $5, you will have a wealth distribution consisting of $295 if the ticket is a winner, and $ if it is not a winner. The original endowment of probabilities of wealth in different circumstances has been changed by the purchase of the lottery ticket. Let us examine this point in more detail. In this discussion we’ll restrict ourselves to examining monetary gambles for convenience of exposition. Of course, it is not money alone that mat- ters; it is the consumption that money can buy that is the ultimate “good” being chosen. The same principles apply to gambles over goods, but re- stricting ourselves to monetary outcomes makes things simpler. Second, we will restrict ourselves to very simple situations where there are only a few possible outcomes. Again, this is only for reasons of simplicity. Above we described the case of gambling in a lottery; here we’ll consider the case of insurance. Suppose that an individual initially has $35, worth of assets, but there is a possibility that he may lose $10,000. For example, his car may be stolen, or a storm may damage his house. Suppose that the probability of this event happening is p = .01. Then the probability distribution the person is facing is a 1 percent probability of having $25, of assets, and a 99 percent probability of having $35,000. Insurance offers a way to change this probability distribution. Suppose that there is an insurance contract that will pay the person $100 if the loss occurs in exchange for a $1 premium. Of course the premium must be paid whether or not the loss occurs. If the person decides to purchase $10, dollars of insurance, it will cost him $100. In this case he will have a 1 percent chance of having $34,900 ($35,000 of other assets − $10,000 loss + $10,000 payment from the insurance payment – $100 insurance premium) and a 99 percent chance of having $34,900 ($35,000 of assets − $100 in- surance premium). Thus the consumer ends up with the same wealth no matter what happens. He is now fully insured against loss. In general, if this person purchases K dollars of insurance and has to pay a premium γK, then he will face the gamble:^1
probability .01 of getting $25, 000 + K − γK
(^1) The Greek letter γ, gamma, is pronounced “gam-ma.”
220 UNCERTAINTY (Ch. 12)
Let’s describe the insurance purchase in terms of the indifference-curve analysis we’ve been using. The two states of nature are the event that the loss occurs and the event that it doesn’t. The contingent consumptions are the values of how much money you would have in each circumstance. We can plot this on a graph as in Figure 12.1.
$25,000 + K – γ K Cb
$35,000 – γ K
Endowment
Choice
Slope = – γ 1 – γ
Cg
$25,
$35,
Figure
Insurance. The budget line associated with the purchase of insurance. The insurance premium γ allows us to give up some consumption in the good outcome (Cg ) in order to have more consumption in the bad outcome (Cb).
Your endowment of contingent consumption is $25,000 in the “bad” state—if the loss occurs—and $35,000 in the “good” state—if it doesn’t occur. Insurance offers you a way to move away from this endowment point. If you purchase K dollars’ worth of insurance, you give up γK dol- lars of consumption possibilities in the good state in exchange for K − γK dollars of consumption possibilities in the bad state. Thus the consumption you lose in the good state, divided by the extra consumption you gain in the bad state, is ΔCg ΔCb
γK K − γK
γ 1 − γ
This is the slope of the budget line through your endowment. It is just as if the price of consumption in the good state is 1 − γ and the price in the bad state is γ.
CONTINGENT CONSUMPTION 221
We can draw in the indifference curves that a person might have for con- tingent consumption. Here again it is very natural for indifference curves to have a convex shape: this means that the person would rather have a constant amount of consumption in each state than a large amount in one state and a low amount in the other. Given the indifference curves for consumption in each state of nature, we can look at the choice of how much insurance to purchase. As usual, this will be characterized by a tangency condition: the marginal rate of substitution between consumption in each state of nature should be equal to the price at which you can trade off consumption in those states. Of course, once we have a model of optimal choice, we can apply all of the machinery developed in early chapters to its analysis. We can examine how the demand for insurance changes as the price of insurance changes, as the wealth of the consumer changes, and so on. The theory of consumer behavior is perfectly adequate to model behavior under uncertainty as well as certainty.
EXAMPLE: Catastrophe Bonds
We have seen that insurance is a way to transfer wealth from good states of nature to bad states of nature. Of course there are two sides to these transactions: those who buy insurance and those who sell it. Here we focus on the sell side of insurance. The sell side of the insurance market is divided into a retail component, which deals directly with end buyers, and a wholesale component, in which insurers sell risks to other parties. The wholesale part of the market is known as the reinsurance market. Typically, the reinsurance market has relied on large investors such as pension funds to provide financial backing for risks. However, some rein- surers rely on large individual investors. Lloyd’s of London, one of the most famous reinsurance consortia, generally uses private investors. Recently, the reinsurance industry has been experimenting with catas- trophe bonds, which, according to some, are a more flexible way to pro- vide reinsurance. These bonds, generally sold to large institutions, have typically been tied to natural disasters, like earthquakes or hurricanes. A financial intermediary, such as a reinsurance company or an invest- ment bank, issues a bond tied to a particular insurable event, such as an earthquake involving, say, at least $500 million in insurance claims. If there is no earthquake, investors are paid a generous interest rate. But if the earthquake occurs and the claims exceed the amount specified in the bond, investors sacrifice their principal and interest. Catastrophe bonds have some attractive features. They can spread risks widely and can be subdivided indefinitely, allowing each investor to bear
EXPECTED UTILITY 223
consumption by the probability that it will occur. This gives us a utility function of the form
u(c 1 , c 2 , π 1 , π 2 ) = π 1 c 1 + π 2 c 2.
In the context of uncertainty, this kind of expression is known as the ex- pected value. It is just the average level of consumption that you would get. Another example of a utility function that might be used to examine choice under uncertainty is the Cobb–Douglas utility function:
u(c 1 , c 2 , π, 1 − π) = cπ 1 c^12 − π.
Here the utility attached to any combination of consumption bundles de- pends on the pattern of consumption in a nonlinear way. As usual, we can take a monotonic transformation of utility and still represent the same preferences. It turns out that the logarithm of the Cobb-Douglas utility will be very convenient in what follows. This will give us a utility function of the form
ln u(c 1 , c 2 , π 1 , π 2 ) = π 1 ln c 1 + π 2 ln c 2.
12.3 Expected Utility
One particularly convenient form that the utility function might take is the following: u(c 1 , c 2 , π 1 , π 2 ) = π 1 v(c 1 ) + π 2 v(c 2 ).
This says that utility can be written as a weighted sum of some function of consumption in each state, v(c 1 ) and v(c 2 ), where the weights are given by the probabilities π 1 and π 2. Two examples of this were given above. The perfect substitutes, or expected value utility function, had this form where v(c) = c. The Cobb- Douglas didn’t have this form originally, but when we expressed it in terms of logs, it had the linear form with v(c) = ln c. If one of the states is certain, so that π 1 = 1, say, then v(c 1 ) is the utility of certain consumption in state 1. Similarly, if π 2 = 1, v(c 2 ) is the utility of consumption in state 2. Thus the expression
π 1 v(c 1 ) + π 2 v(c 2 )
represents the average utility, or the expected utility, of the pattern of consumption (c 1 , c 2 ).
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For this reason, we refer to a utility function with the particular form described here as an expected utility function, or, sometimes, a von Neumann-Morgenstern utility function.^2 When we say that a consumer’s preferences can be represented by an expected utility function, or that the consumer’s preferences have the ex- pected utility property, we mean that we can choose a utility function that has the additive form described above. Of course we could also choose a dif- ferent form; any monotonic transformation of an expected utility function is a utility function that describes the same preferences. But the additive form representation turns out to be especially convenient. If the consumer’s preferences are described by π 1 ln c 1 + π 2 ln c 2 they will also be described by cπ 1 1 cπ 2 2. But the latter representation does not have the expected utility property, while the former does. On the other hand, the expected utility function can be subjected to some kinds of monotonic transformation and still have the expected utility property. We say that a function v(u) is a positive affine transfor- mation if it can be written in the form: v(u) = au + b where a > 0. A positive affine transformation simply means multiplying by a positive num- ber and adding a constant. It turns out that if you subject an expected utility function to a positive affine transformation, it not only represents the same preferences (this is obvious since an affine transformation is just a special kind of monotonic transformation) but it also still has the expected utility property. Economists say that an expected utility function is “unique up to an affine transformation.” This just means that you can apply an affine trans- formation to it and get another expected utility function that represents the same preferences. But any other kind of transformation will destroy the expected utility property.
12.4 Why Expected Utility Is Reasonable
The expected utility representation is a convenient one, but is it a rea- sonable one? Why would we think that preferences over uncertain choices would have the particular structure implied by the expected utility func- tion? As it turns out there are compelling reasons why expected utility is a reasonable objective for choice problems in the face of uncertainty. The fact that outcomes of the random choice are consumption goods that will be consumed in different circumstances means that ultimately only one of those outcomes is actually going to occur. Either your house
(^2) John von Neumann was one of the major figures in mathematics in the twentieth century. He also contributed several important insights to physics, computer science, and economic theory. Oscar Morgenstern was an economist at Princeton who, along with von Neumann, helped to develop mathematical game theory.
226 UNCERTAINTY (Ch. 12)
nature materialize, then if the independence assumption alluded to above is satisfied, the utility function must take the form
U (c 1 , c 2 , c 3 ) = π 1 u(c 1 ) + π 2 u(c 2 ) + π 3 u(c 3 ).
This is what we have called an expected utility function. Note that the expected utility function does indeed satisfy the property that the marginal rate of substitution between two goods is independent of how much there is of the third good. The marginal rate of substitution between goods 1 and 2, say, takes the form
MRS 12 =^ −^
ΔU (c 1 , c 2 , c 3 )/Δc 1 ΔU (c 1 , c 2 , c 3 )/Δc 2
= −
π 1 Δu(c 1 )/Δc 1 π 2 Δu(c 2 )/Δc 2
This MRS depends only on how much you have of goods 1 and 2, not how much you have of good 3.
12.5 Risk Aversion
We claimed above that the expected utility function had some very con- venient properties for analyzing choice under uncertainty. In this section we’ll give an example of this. Let’s apply the expected utility framework to a simple choice problem. Suppose that a consumer currently has $10 of wealth and is contemplating a gamble that gives him a 50 percent probability of winning $5 and a 50 percent probability of losing $5. His wealth will therefore be random: he has a 50 percent probability of ending up with $5 and a 50 percent probability of ending up with $15. The expected value of his wealth is $10, and the expected utility is
1 2
u($15) +
u($5).
This is depicted in Figure 12.2. The expected utility of wealth is the average of the two numbers u($15) and u($5), labeled. 5 u(5) +. 5 u(15) in the graph. We have also depicted the utility of the expected value of wealth, which is labeled u($10). Note that in this diagram the expected utility of wealth is less than the utility of the expected wealth. That is,
u
= u (10) >
u (15) +
u (5).
RISK AVERSION 227
UTILITY
u (15) u (10) .5 u (5) + .5 u (15)
u (5)
u (wealth)
5 10 15 WEALTH
Risk aversion. For a risk-averse consumer the utility of the expected value of wealth, u(10), is greater than the expected utility of wealth,. 5 u(5) +. 5 u(15).
Figure
In this case we say that the consumer is risk averse since he prefers to have the expected value of his wealth rather than face the gamble. Of course, it could happen that the preferences of the consumer were such that he prefers a a random distribution of wealth to its expected value, in which case we say that the consumer is a risk lover. An example is given in Figure 12.3. Note the difference between Figures 12.2 and 12.3. The risk-averse con- sumer has a concave utility function—its slope gets flatter as wealth is in- creased. The risk-loving consumer has a convex utility function—its slope gets steeper as wealth increases. Thus the curvature of the utility function measures the consumer’s attitude toward risk. In general, the more con- cave the utility function, the more risk averse the consumer will be, and the more convex the utility function, the more risk loving the consumer will be. The intermediate case is that of a linear utility function. Here the con- sumer is risk neutral: the expected utility of wealth is the utility of its expected value. In this case the consumer doesn’t care about the riskiness of his wealth at all—only about its expected value.
EXAMPLE: The Demand for Insurance
Let’s apply the expected utility structure to the demand for insurance that we considered earlier. Recall that in that example the person had a wealth
RISK AVERSION 229
probability (1 − π) they pay out nothing. No matter what happens, they collect the premium γK. Then the expected profit, P , of the insurance company is
P = γK − πK − (1 − π) · 0 = γK − πK.
Let us suppose that on the average the insurance company just breaks even on the contract. That is, they offer insurance at a “fair” rate, where “fair” means that the expected value of the insurance is just equal to its cost. Then we have
P = γK − πK = 0,
which implies that γ = π. Inserting this into equation (12.1) we have
πΔu(c 2 )/Δc 2 (1 − π)Δu(c 1 )/Δc 1
π 1 − π
Canceling the π’s leaves us with the condition that the optimal amount of insurance must satisfy Δu(c 1 ) Δc 1
Δu(c 2 ) Δc 2
This equation says that the marginal utility of an extra dollar of income if the loss occurs should be equal to the marginal utility of an extra dollar of income if the loss doesn’t occur. Let us suppose that the consumer is risk averse, so that his marginal utility of money is declining as the amount of money he has increases. Then if c 1 > c 2 , the marginal utility at c 1 would be less than the marginal utility at c 2 , and vice versa. Furthermore, if the marginal utilities of income are equal at c 1 and c 2 , as they are in equation (12.2), then we must have c 1 = c 2. Applying the formulas for c 1 and c 2 , we find
35 , 000 − γK = 25, 000 + K − γK,
which implies that K = $10, 000. This means that when given a chance to buy insurance at a “fair” premium, a risk-averse consumer will always choose to fully insure. This happens because the utility of wealth in each state depends only on the total amount of wealth the consumer has in that state—and not what he might have in some other state—so that if the total amounts of wealth the consumer has in each state are equal, the marginal utilities of wealth must be equal as well. To sum up: if the consumer is a risk-averse, expected utility maximizer and if he is offered fair insurance against a loss, then he will optimally choose to fully insure.
