# Why is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013)

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1 Why is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013) Introduction The United States government is, to a rough approximation, an insurance company with an army. 1 That is because the majority of federal government spending represents some form of social insurance. Rising health care costs and an aging population are expected to make the share of government spending devoted to social insurance in the U.S. grow rapidly in the future, and social insurance is already a much larger share of government in essentially every other rich country than it is in the U.S. So to understand modern political economy, it is absolutely critical to have a solid understanding of why insurance is good. That insurance has value is a non-obvious insight, because on average, people lose money when they buy insurance, due to the need to cover the costs of the insurance company. So why would people ever voluntarily buy insurance? How could it possibly make them better off? Let s consider a simple example that gets the idea across. Set-Up of the Example Suppose you face a risky situation. To keep the math simple, suppose there is a 50% probability that you will have good luck, in which case your consumption will be \$90,000, and a 50% probability that you will have bad luck, in which case your consumption will be \$10,000. You can think of the difference as representing some catastrophic expense that occurs randomly, such as a large health expenditure, in which case by consumption we mean consumption aside from the catastrophic expense. Or you can think of it as a drop in consumption arising, for example, from an unexpected job loss. Further suppose that your utility equals the square root of your consumption, which is a mathematically convenient function that exhibits diminishing marginal utility of consumption. So to summarize: Utility = U(C) = C = C 1/2. Note that U(C) means utility as a function of consumption. Consumption if good luck, Cg = \$90,000 Consumption if bad luck, Cb = \$10,000 Probability of good luck, Pg = 50% Probability of bad luck, Pb = 50% The expected value of something is just its value on average. You can calculate the expected value of something by multiplying each possible value of the variable by its probability, and then summing up. In general, the symbol for expected value of something is E(something). 1 The original source of this quote is Peter Fisher, from when he was Undersecretary of the Treasury, in 2002 < 1

2 The average, or expected, value of consumption, E(C), when facing this risky situation, is: E(C) = Pb Cb + Pg Cg = 0.5 \$10, \$90,000 = \$50,000. We ll denote this as E(C)NO INSURANCE, to remind us that this is your expected value of consumption when you do not have insurance. The average, or expected, value of utility, E(U), when facing this risky situation, is: E(U) = Pb U(Cb) + Pg U(Cg) = , , 000 = 200 utils. We ll denote this as E(U)NO INSURANCE, to remind us that this is your expected value of utility when you do not have insurance. A rational person should behave so as to maximize his or her own expected value of utility, not his or her own expected value consumption. Utility is your level of happiness. That s what you care about. Consumption is only instrumental to utility (happiness). So you don t care directly about the dollar value of consumption, but instead you care about the amount of utility provided by that dollar value of consumption, and you should behave so as to maximize expected utility. Why Risk is Bad Diminishing marginal utility implies that on average, your expected level of happiness (utility) would be higher if you could somehow ensure that you would always get the expected value of your consumption with certainty, instead of having consumption that was much lower than its expected value in certain bad luck states of the world that might occur, and much higher than its expected value in other good luck states of the world that might occur. This is true because, if there is diminishing marginal utility from consumption, the gain in utility from increasing consumption above its expected value in the good luck state of the world is smaller than the loss of utility from reducing consumption below its expected value by the same dollar amount in the bad luck state of the world. Figure 1 below depicts the scenario described in the example above, and demonstrates the conclusion that risk is bad in a more systematic way. If you could somehow maintain the expected value of your consumption of \$50,000 with certainty, then your utility would be U(50,000) = 50, 000 = 224 utils. That is better than the 200 utils of utility that you expect to get, on average, from the risky situation where half the time you consume \$10,000 (yielding a utility of 100) and the other half of the time you consume \$90,000 (yielding a utility of 300). Figure 1 illustrates this as follows. Compared to situation where you get the expected value of your consumption (\$50,000) with certainty, which yields a utility of 224, the gain in utility from an increase in consumption from \$50,000 to \$90,000 is only ( ) = 76 utils, whereas the loss in utility from an equal-dollar decrease in consumption, from \$50,000 to \$10,000, is a much larger ( ) = 124 utils. Going up by \$40,000 relative to a 2

3 given starting point increases your happiness by much less than going down by \$40,000 from the same starting point reduces your happiness. That will be true as long as your utility function exhibits diminishing marginal utility of consumption. A person whose utility function exhibits diminishing marginal utility of consumption will thus dislike risk, will be willing to pay to avoid it, and will demand compensation to accept it we call such a person risk averse. Note that the argument for why risk is bad relies on the same principle that is behind the utilitarian argument for why redistribution from rich to poor can increase the sum of utilities in society. In our stylized example, you prefer having \$50,000 for sure to having \$10,000 with 50 percent probability and \$90,000 with 50 percent probability. If you could somehow redistribute \$40,000 away from yourself in the possible state of the world where you have \$90,000, and send it to yourself in the possible state of the world where you only have \$10,000, the probabilityweighted sum of your utilities across the two states of the world (that is, your expected utility) would go up. That is more or less what insurance does. The difference from redistribution is that you are willing to enter into an insurance contract voluntarily, because you don t know in advance whether the good luck or bad luck state of the world will apply to you. As a result, under certain conditions, private insurance companies can make money supplying insurance. 3

8 \$50,000 - \$41,000 = \$9,000. As long as the administrative cost per customer is lower than the difference between MWTP and the marginal cost of actuarially fair full insurance, there is positive economic surplus to be had from the insurance contract, and room for a deal to be made between consumers and producers that makes both better off. Summary and Conclusion To summarize, the following steps are involved in calculating economic surplus from insurance, and in figuring out who will buy insurance when it is offered at a particular price. 1. Calculate a person s certainty equivalent consumption in the absence of insurance. 2. Calculate the person s maximum willingness-to-pay for the insurance. This is the price of insurance that would make the guaranteed consumption with the insurance exactly equal to the certainty equivalent consumption without the insurance. Remember that to find guaranteed consumption with the insurance, you need to subtract off the price paid for the insurance from the level of consumption the insurance would guarantee the person if he or she got the insurance for free. 3. Calculate the marginal cost and the price of the insurance. The marginal cost equals the administrative cost per customer, plus the expected value of the benefit payout per customer. In a competitive market, the insurance company sets the price of the insurance equal to marginal cost. a. If there are multiple risk types, and the insurance company can tell them apart, then the insurance will be sold at a different price to each customer, with price equal to marginal cost for that type of customer. b. If there are multiple risk types and the insurance company cannot tell them apart, then each insurance company will charge a price equal to the average marginal cost of all the customers it expects to attract. If at first it gets this wrong, it will eventually adjust its premium to reflect the marginal cost of the customers it actually attracts, so that it can break even. 4. An individual will voluntarily buy insurance if and only if maximum willingness-to-pay is greater than or equal to the price at which he or she is offered insurance. 5. Consumer surplus for each customer, if he or she buys the insurance, is his or her maximum willingness-to-pay minus price. 6. Producer surplus from each customer, if he or she buys the insurance, is price minus marginal cost. 8

9 7. Economic surplus associated with each customer is the sum of consumer and producer surplus. 8. Deadweight loss is the amount of economic surplus lost if a market failure (such as asymmetric information) prevents a mutually beneficial insurance contract from being purchased. The basic insights are easier to see when the probability of bad luck is 50%, but do not depend on it. The key point is that as long as administrative costs are low enough and firms can charge a premium to each customer that reflects the customer s true level of risk, then the insurance will be mutually beneficial to customer and insurance company, and will increase both utility and economic surplus in society relative to the absence of insurance. Diminishing marginal utility is what makes this work. When there is diminishing marginal utility, moving consumption away from yourself in high-consumption states of the world, where marginal utility is low, and towards yourself in low-consumption states of the world, where marginal utility is high, can increase the expected value of your utility. So you are voluntarily willing to do things that make that happen. 9

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