Choice of an action in Savage’s theory is frequently interpreted as choice of a lottery over wealth (or income, the difference between the two doesn’t matter here). Suppose DM cares about wealth (we write that as w) and suppose that choice of an action can be read as choice of a lottery. The u-function is then defined over wealth, that is, u(w) is the u-function.
In Savage’s theory u is an increasing function of w. To say that DM is risk averse over all portfolio choices is to say that u(w) is (strictly) concave, that is, marginal utility is a declining function of w. Formally, that means uˊ(w) – the first derivative of u - is positive and uˊˊ(w) – the second derivate of u – is negative (Figure 5.1).
Let wi be DM’s wealth if the state of Nature is i. As there are N states of Nature, a lottery is an N-vector (w1,...,wi,...,wN). Intuitively it is apparent that the degree of risk aversion displayed
172 Temperature scales are cardinal. The Fahrenheit and Centigrade scales are related in the way described in the text. Conversion from the Centigrade scale to the Fahrenheit scale requires multiplying by 9/5 (that is the β) and adding 32 (that is the γ).
Chapter 5: Risk and Uncertainty
by DM is reflected in the curvature of the u-function, a convenient measure of which is the elasticity of marginal u with respect to w. Denoting the elasticity of marginal u by η, we have η = η(w) = - wuˊˊ(w)/uˊ(w) > 0. For obvious reasons η(w) is also known as the measure of DM’s relative risk aversion (Pratt, 1964; Arrow, 1965).
In the economics of climate change it is near-universal practice to assume that relative risk aversion, η, is independent of wealth (Chapter 10). The family of u-functions for which η is a positive constant has the form
u(w) = w(1-η)/(1-η), η≠ 1,
u(w) = log w, η = 1 (B5.2.1)
The larger is η, the greater is the curvature of u-function. Notice also that u(w) is bounded above but unbounded below if η > 1, whereas u(w) is bounded below but unbounded above if η < 1. If η = 1, u(w) is unbounded at both ‘ends.’ The asymptotic properties of the u-function in equation (B5.1.1) are its weakness, for Savage’s theory yields bounded u-functions. Annex 5.1 reveals that the use of unbounded u-functions gives rise to paradoxes, known as St. Petersburg Paradoxes.
So as to confine attention to bounded u-functions while retaining the empirical finding that, generally speaking, η is less than 1 for small w but greater than 1 for large w, Arrow (1965) proposed the use of u-functions that are bounded, and for which η is (i) an increasing function of w and (ii) less than 1 for small w and greater than 1 for large w. Condition (ii) implies that η = 1 at some intermediate value of w. Here is an example of a u-function with these properties:
u(w) = 1 – e-λw (B5.2.2)
The measure of relative risk aversion for the u-function in equation (B5.2.2) is η(w) = λw.
As an example of bounded u-function equation (B5.2.2) is attractive, because it is built on a single parameter, λ. But the functional form is not without a problem. To reflect the fact that to be destitute is the ultimate calamity for a person, it makes sense to postulate that uˊ(w) ⟶
∞ as w ⟶ 0. In the example here, uˊ(w) tends instead to λ.
Arrow and Priebsch (2014) have shown that the requirements that (i) u(w) is bounded and (ii) uˊ(w) ⟶∞ as w ⟶ 0 cannot be integrated to yield a closed functional form for u. As the authors show, that does not prevent DM to study the economics of risk and uncertainty, but it is harder work.
In certain applications, it makes sense to extend Savage’s theory and work with u-functions that are explicit functions of the state of Nature. In studying optimum expenditures on health, for example, one may take the state of a person’s health, labelled as h, as a state of Nature and express the u-function as u(w,h). The idea here is that a person’s needs depend on his health.
5.2 Independent vs. Correlated Risks
It is useful to discuss extreme cases, and so we distinguish between risks that are independent across individuals from those that are perfectly positively correlated across individuals.173 We begin with the case of independent risks. We then study perfectly (positively) correlated risks.
173 Negative correlation in the risks people face is of course a possibility. Firm 1 could hold a patent on the genetic material of plant A, while firm 2 holds the genetic material of plant B and it is commonly known that one and only one of them is a cure for a disease, but not which.
Chapter 5: Risk and Uncertainty
5.2.1 Ideal Insurance Markets
The risks people face from being involved in road accidents are independent of one another, which is why insurance markets for road accidents are able to flourish. People pool their risks in the insurance market; in turn insurance firms diversify their risks by selling accident insurance to large populations. One way to interpret insurance markets is to imagine that all who purchase insurance mutually insure one another and use insurance companies as intermediaries. Under ideal market conditions competition among insurance firms drives the premium for a risk category down to the expected loss from an accident in that category. It would be as though people deposited their expected loss from a risk in that category into a common fund from which they were entitled to take what they had lost in case they had an accident.
Consider for simplicity the case where everyone belongs to the same risk category. Let w denote a person’s wealth and let w* be his wealth if he was not to suffer from an accident (Figure 5.1).
If π is the probability that he will suffer from an accident, and m is the monetary loss from the accident, then πm would be the premium. m may be large, the accident could even be a catastrophe for the person, but if π is sufficiently small, the premium would be small relative to w*.174 More importantly, because the risks are independent of one another, people would be able to insure themselves fully against personal accidents at that price (at least in theory!).
The reason they would be able to do that is that if the population were large, the Law of Large Numbers in probability theory says everyone could be ‘guaranteed’ the wealth (w*-πm) if they mutually insured one another. As πm is small relative to w*, people would be protected (almost) fully against loss. The insurance market in principle does that; and one says the market pools risks.175
This, in very broad terms, is the theory underlying ideal insurance markets. In practice any number of features of the world muddy the account. For example, in many places people do not have access to insurance markets even for personal risks such as illness and death in the family. In the absence of markets, rural households in poor countries form networks for pooling personal risks (Chapter 6). Village numbers being small, households cannot take advantage of the Law of Large Numbers, so they are able to insure one another only partially. But networks have two advantages insurance markets do not enjoy. Proximity among households and the fact they know one another well mean that rural insurance schemes suffer far less than insurance markets from ‘moral hazard’ and ‘adverse selection’.
Moral hazard is the inability of insurance companies to ensure that the individual purchasing insurance takes due diligence. The point here is that companies are unable to observe insurees’
actions. Without due diligence in our example the risk the insuree will have an accident would exceed π. Moral hazard is a ubiquitous problem in insurance markets. By introducing suitable incentives insurance companies reduce the moral hazard they would otherwise face. In automobile insurance, for example, companies reduce moral hazard by requiring applicants to produce an acceptable certificate each year that says their car is roadworthy. Moral hazard also lies behind the practice of lowering the premium if no claim had been made in the previous year. For property insurance against burglary, moral hazard is reduced by a requirement that the insuree installs safety features in his home (burglar alarm, sturdy locks, and so forth).
The other ubiquitous feature of insurance markets is adverse selection, which is the inability of an insurance company to fully determine an insuree’s relevant characteristics. In the case of automobile insurance, insurance companies remedy this partially by requiring applicants to produce evidence that they are safe drivers (e.g. they enjoy good eyesight). Thus, while moral
174 In Annex 5.2 we comment on the case where the risks are perfectly positively corrected, π is small, but the collective loss would be a catastrophe, meaning that πm is huge, maybe even unboundedly large. The expected utility theory will not hold in this case.
175 Malinvaud (1972) is a standard source for an account of the economics of insurance.
Chapter 5: Risk and Uncertainty
hazard points to the problems insurance companies face in confirming that insurees do what is required of them under the terms of the contract, adverse selection points to the problems companies face in confirming that insurees are what they claim to be (e.g. regarding their risk category).
Moral hazard and adverse selection give rise to externalities, which are the unaccounted-for consequences for others of actions taken by someone. Externalities, which are discussed in Chapters 6-9, are a pervasive feature in the economics of biodiversity. They reflect a class of imperfections in the allocation of resources, which in the present context are imperfections in the production and distribution of risk. In what follows, we do not formally model either moral hazard or adverse selection, but we do point out on appropriate occasions the steps insurers take to reduce them. We therefore interpret π to be the average risk across the population.
5.2.2 Fully Correlated Risks
The examples that were studied in Chapters 2-4 tell us that the risks associated with biodiversity loss and global climate change are positively, often perfectly, correlated.176 Losses from floods and tropical cyclones can significantly affect a community’s wealth, particularly in low income countries. In extreme cases, communities can be made destitute by one such calamitous event (Chapters 14 and 17). Earthquakes are another class of natural occurrences. Because the risks are perfectly correlated, they are called common risks. The Law of Large Numbers does not apply to them. If the community is an entire country and insurance companies are unable to diversify their risks beyond the country’s borders, they would not be able to offer cover against a common risk.
Because there is no scope for insurance, each person in the country would have to bear his entire risk. Suppose w* would be a person’s wealth if the catastrophe were not to occur (or at least not for many years from now) but would be (w*-m) if it did occur. If u(y) is the person’s utility function, his expected utility would be [(1-π)u(w*) + πm)], which is less than u(w*-πm), because the person is risk-averse. In Figure 5.1, income w is drawn on the horizontal axis and utility u is drawn on the vertical axis. u is an increasing function of w and, because DM respects people’s aversion to risk, its shape is concave. If the person was not to suffer the loss, his utility would be u(w*), which in Figure 5.1 is BB’. If on the other hand he was to suffer the loss, his utility would be u(w*-m), which is AA’. In Box 5.2 we hold constant the potential loss m and the probability π that the loss will occur so as to study two extreme cases: (i) the risks are independent and people are fully insured against them; and (ii) the risks are common and people are fully exposed to them.