I had an email exchange with one of your classmates this weekend about expected utility and the certainty equivalent of wealth in the context of solving problem 5 on the problem set, so I thought that it might be a good idea to share with the class generally what I shared with him.
The utility function tells you (among other things) how to find the certainty equivalent. For example, consider the diagram below involving the toss of a fair coin (note that a “fair” coin implies equal probabilities of heads and tails). If heads comes up, wealth is W1 and if tails comes up wealth is W2. Expected wealth is E(W) = .5(W1) + .5(W2). Expected utility from this coin toss is E(U(W)) = .5U(W1) + .5U(W2). If given a choice between a fair coin toss which pays off E(W) with some risk, compared with getting the sum of E(W) with no risk, the person depicted here prefers the latter choice, because the utility of receiving E(W) without having to bear any risk (U(E(W)) is greater than the utility of receiving E(W) and having to bear risk (E(U(W)). Note that since U(E(W)) > E(U(W)), this person is risk averse, because given the choice between a sum of money with no risk and the expected value of that same sum with some risk, she prefers the former.
Since U(WCE) = E(U(W)) in this diagram, we know that WCE is the certainty equivalent; it corresponds to the level of “safe” wealth which provides the same level of utility as receiving the expected value of the coin toss E(W). Finding WCE for any given problem involves applying basic algebra to the utility function that is specified. For example, suppose utility is logarithmic; i.e., U(W) = ln W. Then U(WCE) = ln WCE —> WCE = exp(U(WCE)) = exp(E(U(W)) (note that exp(x) = ex). Similarly, if U(W) = W.5, then U(WCE) = WCE.5 —> WCE = U(WCE)2 = E(U(W))2, and so on.
Problem 5 is similar to the problem that I outlined above, except it involves an unfair coin toss; in this case, heads (getting into an accident) is 20%, whereas tails (not getting into an accident) is 80%. A particularly interesting aspect of problem 5 is that because of risk aversion, the insurance company can offer unfair bets to consumers. Think about it. If the consumer were risk neutral, then she would be indifferent between having an insurer bear her risk for an “actuarially fair” premium of $60 and bearing this risk herself. However, since she is risk averse, if actuarially fair insurance were available, she would (without any reservation whatsoever) fully insure (this principle – full insurance at an actuarially fair price – is commonly referred to as the “Bernoulli principle”, in honor of Daniel Bernoulli). Furthermore, since she is risk averse, she is willing to pay more than $60 in order to avoid having to bear this risk. The reason why a transaction like this will occur is because the cost of risk bearing is lower for the insurer than it is for the consumer; specifically, the insurer is in a position, unlike the consumer, to take advantage of the law of large numbers. Therefore, “actuarially unfair” insurance (i.e., insurance costing more than $60) actually enhances social welfare, while enabling the insurer to earn some profit. Social welfare is enhanced because risk bearing is allocated to an economic institution (the insurance company) which has a comparative advantage in risk bearing. Similar points can be made concerning allocating capital in the economy via financial markets – there, investors play the insurer’s role in terms of managing risk at a lower cost than the entrepreneur.
Anyway, I hope this short essay is helpful; it provides a useful segue into the Chapter 14 topic, which involves figuring out how to parameterize compensation contracts for corporate executives in a way that aligns investor/manager incentives.