Why insurance works better with some adverse selection

Adverse selection, a process whereby low-risk individuals drop out of the insurance pool, leaving only high-risk individuals, arises when the individuals purchasing insurance have better information regarding their risk status than does the insurer. […] In the limit, adverse selection can make insurance markets unsustainable. Even short of the market disappearing altogether… The market cannot offer a full set of insurance contracts, reducing allocative efficiency.

The story summarised above (by Jeremiah Hurley) is familiar to all health economists. Adverse selection is generally understood to be a universal problem for efficiency in health insurance (and indeed all insurance), which should always be avoided or minimised, or else traded off against other objectives of equity. In my book, Loss Coverage: Why Insurance Works Better with Some Adverse Selection, I put forward a contrary argument that a modest degree of adverse selection in insurance can increase efficiency.

My argument depends on two departures from canonical models of insurance, both realistic. First, I assume that not all individuals will buy insurance when it is risk-rated; this is justified by observation of extant markets (e.g. around 10% of the US population has no health insurance, and around 50% have no life insurance). Second, my criterion of efficiency is based not on Pareto optimality (unsatisfactory because it says so little) or utilities (unsatisfactory because always unobservable), but on ‘loss coverage.’

In its simplest form, loss coverage is the expected fraction of the population’s losses which is compensated by insurance.

Since the purpose of insurance is to compensate the population’s losses, I argue that higher loss coverage is more efficient than lower loss coverage. Under this criterion, insurance of one high risk will contribute more to efficiency than insurance of one low risk. This is intuitively reasonable: higher risks are those who most need insurance!

If this intuition is accepted, the orthodox arguments about adverse selection seem to overlook one point. True, adverse selection leads to a higher average price for insurance and a fall in numbers of individuals insured. But it also leads to a shift in coverage towards higher risks (those who need insurance most). If this shift in coverage is large enough, it can more than outweigh the fall in numbers insured, so that loss coverage is increased.

My argument can be illustrated by the following toy example. The numbers are simplified and exaggerated for clarity, but the underlying argument is quite general.

Consider a population of just ten risks (say lives), with three alternative scenarios for insurance risk classification: risk-differentiated premiums, pooled premiums (with some adverse selection), and pooled premiums (with severe adverse selection). Assume that all losses and insurance cover are for unit amounts (this simplifies the discussion, but it is not necessary).

The three scenarios are represented in the three panels of the illustration. Each ‘H’ represents one higher risk and each ‘L’ represents one lower risk. The population has the typical predominance of lower risks: a lower risk-group of eight risks each with probability of loss 0.01, and a higher risk-group of two risks each with probability of loss 0.04.

In Scenario 1, risk-differentiated premiums (actuarially fair premiums) are charged. The demand response of each risk-group to an actuarially fair price is the same: exactly half the members of each risk-group buy insurance. The shading shows that a total of five risks buy insurance.

Scenario 1


The weighted average of the premiums paid is (4 x 0.01 +1 x 0.04)/5 = 0.016. Since higher and lower risks are insured in the same proportions as they exist in the population, there is no adverse selection.

Exactly half the population’s expected losses are compensated by insurance. I describe this as ‘loss coverage’ of 50%. (The calculation is (4 x 0.01 + 1x 0.04) / (8 x 0.01 + 2 x 0.04) = 0.50.)

In Scenario 2, risk classification has been banned, and so insurers have to charge a common pooled premium to both higher and lower risks. Higher risks buy more insurance, and lower risks buy less (adverse selection). The pooled premium is set as the weighted average of the true risks, so that expected profits on low risks exactly offset expected losses on high risks. This weighted average premium is (1 x 0.01 +2 x 0.04)/3 = 0.03. The shading symbolises that that three risks (compared with five previously) buy insurance.

Scenario 2


Note that the weighted average premium is higher in Scenario 2, and the number of risks insured is lower. These are the essential features of adverse selection, which Scenario 2 accurately and completely represents. But there is a surprise: despite the adverse selection in Scenario 2, the expected losses compensated by insurance for the whole population are now higher. That is, 56% of the population’s expected losses are now compensated by insurance, compared with 50% before. (The calculation is (1 x 0.01 + 2 x 0.04) / (8x 0.01 + 2 x 0.04) = 0.56.)

I argue that Scenario 2, with a higher expected fraction of the population’s losses compensated by insurance – higher loss coverage – is more efficient than Scenario 1. The superiority of Scenario 2 arises not despite adverse selection, but because of adverse selection.

At this point an economist might typically retort that that the lower numbers insured in Scenario 2 compared with Scenario 1 is suggestive of lower efficiency. However, it seems surprising that an arrangement such as Scenario 2, under which more risk is voluntarily traded and more losses are compensated, is always disparaged as less efficient.

A ban on risk classification can also reduce loss coverage, if the adverse selection which the ban induces becomes too severe. This possibility is illustrated in Scenario 3. Adverse selection has progressed to the point where only one higher risk, and no lower risks, buys insurance. The expected losses compensated by insurance for the whole population are now lower. That is, 25% of the population’s expected losses are now compensated by insurance, compared with 50% in Scenario 1, and 56% in Scenario 2. (The calculation is (1 x 0.04) / (8x 0.01 + 2 x 0.04) = 0.25.)

Scenario 3


These scenarios suggest that banning risk classification can increase loss coverage if it induces the `right amount’ of adverse selection (Scenario 2), but reduce loss coverage if it generates `too much’ adverse selection (Scenario 3). Which of Scenario 2 or Scenario 3 actually prevails depends on the demand elasticities of higher and lower risks.

The argument illustrated by the toy example applies broadly. It does not depend on any unusual choice of numbers for the example. The key idea is that loss coverage – and hence, I argue, efficiency – is increased by a modest degree of adverse selection.


7 thoughts on “Why insurance works better with some adverse selection

  1. An interesting thought experiment. Scenario 2 could presumably be argued to be somewhere between an optimal level of risk-sharing and a bit adverse selection that people might choose before knowing their condition. But, I wonder in a market with multiple plans if it would still stand up since high and low risk people will differentially select into more and less generous plans, distorting their prices so that they no longer reflect the marginal costs to individuals leading to an efficiency loss that way.

    Liked by 1 person

  2. Well multiple plans make the argument more complicated. But even in my simple Scenario 2, the price doesn’t reflect the marginal costs for individuals. The high risks are getting their insurance below its marginal cost, and the low risks are getting it above marginal cost. I’m saying that it’s more efficient for price NOT to reflect marginal cost (I realise this is heretical). This depends, of course, on my criterion of efficiency. But under my approach, more risk is being voluntarily traded, and more losses are being compensated; I find it hard to see why that’s bad.


    1. I think my point was more that if you had more than one insurance scheme each paying out a different set of benefits, then the distortions in price by people selecting into the programs may mean there is a net loss in efficiency and/or loss coverage, as you call it. Another issue is that insurance will always be priced too high for a risk neutral individuals as the insurer has to make money. So if the price rises due to adverse selection in a generous scheme, the price may become too high leading to high risk individuals to select into a less generous product raising that price and pricing out low risk individuals. Then, my feeling is that at equilibrium you will end up much more likely with something resembling Scenario 3. It would be interesting to see under what conditions Scenario 2 could be achieved. Presumably you would need some kind of subsidy or price cap (perhaps along the lines of the ACA).


  3. Best place to buy the book is direct from Cambridge University Press http://bit.ly/2q4yUth and enter THOMAS2017 at checkout for a 20% discount. Or if anyone has a half-credible claim to be a potential reviewer, send me an address and I can probably get CUP to send a freebie.


  4. I have not modelled different plans. I can see it might be a salient and realistic issue for healthcare. Absent different plans, there are simple (and in my view often plausible) demand elasticity conditions which guarantee that Scenario 2 will be achieved. For iso-elastic demand, just demand elasticity less than 1. For any downward sloping demand functions, conditions on arc elasticities of demand between pooled and actuarially fair prices. As in this paper http://www.guythomas.org.uk/pdf/IME-Jun17.pdf.


  5. In Scenario 1, your demand response has a lot of underlying assumptions. It could be that the demand response (because the premium is equal to the person’s risk) is for persons to buy insurance with probability 1/2, a defensible assumption but still an assumption (you would also seem to be assuming risk neutrality, whereas most people are risk averse). Or it could be that the risk tolerances of the population are different thus you have some buying insurance and others not. With Scenario 2, the premium is 3 times the L-type’s expected risk. For it to make sense to buy insurance at that premium, he would have to be relatively risk averse (thus you are using different behavioral assumptions relative to Scenario 1). Scenario 3 is subject to the Scenario 1 criticism. This is admittedly a critique of your toy example; your book might be a bit more detailed. I kind of understand why you use loss coverage as a measure of efficiency, but it would also seem that perfect price discrimination might provide a higher loss coverage depending on the preferences of the buyers. It seems even good enough price discrimination in which insurers can reasonably estimate personal risk and give a pretty good guess as to the risk preference of each person would dramatically out perform a market with even the mild adverse selection you describe.


  6. . Hi Jerrod. Thanks for the comments. Responding to each of your points. (1) I am not assuming risk neutrality. As per your second surmise, the demand function can be micro-founded in a distribution of risk preferences across individuals. Details in this.paper http://www.guythomas.org.uk/pdf/IME-Jun17.pdf. I actually view the micro-foundation as optional: for me a more important justification is the observable fact that, for whatever reason, when insurance is risk-rated some people do not buy. (2) The difference in high and low risk levels is exaggerated for clarity. Every part of the argument still works with a smaller difference. (3) I agree with your last two sentences. Depending on demand elasticities, any of full, partial or nil risk classification could maximise loss coverage. This is covered in Chapter 6 of the book.


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