Pictures of adverse selection in an insurance market, with and without death spirals

This is a simple model, with diagrams, of adverse selection in an insurance market. It's mostly for teaching purposes. (Adverse selection is currently very topical in the US, but it's a perennial problem that applies to all forms of insurance markets, and many other markets too.)

I don't know if these diagrams are in any way original. I don't know how other people teach adverse selection, but I would be interested to compare.

Adverse selection depends on asymmetric information. I assume that each individual knows his own risk, but that the insurance company is unable (or is not allowed) to discriminate between individuals and must charge the same premium to all. Since all pay the same premium, the higher risk individuals will be more likely to buy insurance than the lower risk individuals.

The "expected loss" for each individual is equal to the probability of an accident times the loss if an accident does occur. Line up all the individuals along the horizontal axis of the diagram, starting with the highest risk individual (with the highest expected loss) at the left, and ending with the lowest risk individual (with the lowest expected loss) on the right. The blue curve shows the expected loss of the Marginal individual, as a function of the percentage of the population that buys insurance. The green curve shows the Average expected loss of all individuals who buy insurance. The green Average Expected Loss curve will always lie above the blue Marginal Expected Loss curve, because the Marginal Expected Loss is the expected loss that is lowest for all the individuals who buy insurnace.

If we assume that insurance companies are risk-neutral, earn zero profits, and have no costs (like administrative costs) other than the cost of paying out for insurance claims, then the green curve is also the supply curve of insurance. The price of insurance (the insurance premium) must be just sufficient to cover the average expected losses of the individuals who buy insurance. Notice that the supply curve slopes down. The greater the percentage of the population that buys insurance, the lower the average cost per individual of providing that insurance, and the lower the price needs be to cover that cost.

If individuals were risk-neutral, each would only buy insurance if the price were less than his expected loss. The blue curve would then be the demand curve for insurance for risk-neutral individuals. But I will assume that individuals are risk-averse. A risk-averse individual has a Willingness To Pay for insurance (a "demand price") that is greater than his expected loss. The red curve is the demand curve for insurance, and lies everywhere above the blue Marginal Expected Loss curve.

(If the highest risk individual had a 100% probability of having an accident, risk-aversion wouldn't matter for that individual, and the red curve would meet the blue curve where Q=0%. And if the lowest risk individual had a 0% probability of having an accident, risk-aversion wouldn't matter for that individual, and the red curve would meet the blue curve where Q=100%.)

In this model there is one equilibrium where the green supply curve crosses the red demand curve. The premium is P* and a percentage Q* of the population buys insurance at that price.

That equilibrium is not efficient. It fails to maximise consumers' surplus (producers' surplus is zero by assumption). The efficient allocation is where every individual buys insurance. To see this, note that the marginal cost curve is the blue curve, and the marginal benefit curve is the red curve, and marginal benefit exceeds marginal cost for every individual. If insurance companies could price-discriminate then every individual would buy insurance. But price discrimination would require we drop the assumption of asymmetric information. If the government made insurance mandatory for all individuals, the premium would drop to where the green curve cuts the 100% line, but the lower risk individuals would be worse off as a result. (But the winners could in principle compensate the losers.)

Death Spirals.

Now let's change the model to introduce an administrative cost, over and above the cost of paying out for claims. The new brown curve is now the supply curve, and it lies everywhere above the original green supply curve. I have drawn it so that it crosses the red demand curve twice. So there are now two equilibria.

The stable equilibrium is very similar to the equilibrium in the previous diagram. If the insurance company sets the premium a little bit above/below the equilibrium, it will find the premium is above/below the average cost (including administrative costs), and competition will force it to lower/raise the premium back to the equilibrium. So it is indeed a (locally) stable equilibrium.

The unstable equilibrium is quite different. If the insurance company sets the premium above the equilibrium, it loses customers, and average costs rise above the premium, and if it raises the premium even further to cover costs, this just worsens the problem. That's when we get the "adverse selection death spiral", where the insurance market disappears altogether.

(You might say that the insurance company would realise that raising premiums to cover costs would only make matters worse, and that it should cut premium instead. Maybe. But the insurance company may not know whether the equilibrium is stable or unstable, so does not know whether it should raise or lower premiums if premiums do not cover costs. It may not know whether any equilibrium even exists.)

Comments

[Chris Auld]

Great exposition, Nick. Not sure if it's original either, but I don't recall ever seeing the result presented in this form. The usual undergrad level exposition, in health anyways, is the two-type but variable insurance coverage Stiglitz-Rothschild 1976 model.

[Wonks Anonymous]

I was surprised when I read adverse selection was not a big problem even in the auto market, which was what inspired the "market for lemons" theory in the first place:
http://marginalrevolution.com/marginalrevolution/2005/12/adverse_selecti.html
I don't have access to the linked jstor paper though. Is anybody here familiar with the actual data on "lemons"?
And on an only slightly related point, I still find one of Bryan Caplan's post titles, "Adverse Hazard, Moral Selection, Whatever", hilarious.

[Determinant]

It depends on the operating assumptions of the insurer in the case of auto. In Canadian private auto insurance provinces, insurers can use (mostly) whatever information they want to set a premium. Crash history, claims history, infractions, location, etc.

"Lemons", really high risk individuals, are sent to the Facility Association, which is a pool where each insurer who sells auto insurance in the province participates to a stated degree, which is how adverse selection is controlled.

In another example, GEICO, properly Government Employees Insurance Company started because it could offer lower premiums to US Federal Government employees, who had lower crash rates. Eventually the pool was expanded to other occupations with higher crash rates and in the 1970's GEICO went into a death spiral. It had to close units, raise rates and shed customers to regain solvency.

[Nick Rowe]

Wonks: two good links there (and good discussion on them in comments).

Here is the Bryan Caplan one

Here is the Alex Tabarrok one

[Chris Auld]

I wrote a short comment which seems to have disappeared. Assuming a spam filter ate it, it was along the lines-

Nick, great exposition, I'm going to swipe it for undergrad health economics. I don't know if it's original, either, but the usual presentation, in health anyways, is based on the two-type but variable coverage Stiglitz-Rothschild 1976 model, which can get at some different issues but is also more challenging.

Determinant, "lemons" are not high-risk people per se, they're people who are high risk conditional on what the insurer can observe and legally contract over. You can take Nick's analysis as holding for any group of people who look the same to an insurer, e.g. something like: all 25 year old males with two accidents and four speeding tickets in the last year.

[Shangwen]

The death spiral issue in the US is of course related to ACA/Obamacare, and the individual mandate is one of the policies meant to reduce adverse selection. The law has a provision for a component called Risk Corridors to offset year-by-year losses if expected losses are too high. The dilemma here (from a markets perspective) is that these losses to private insurers are made whole through taxation.

For a quick description of the risk corridor, see here.

[Nick Rowe]

Chris: Thanks! I fished your first comment out of the spam filter.

Yes, I should have made it explicit: I'm assuming full insurance with no co-pay percentage and no deductible.

[Nick Rowe]

Thinking about Determinant's GEICO example: it's not a *true* death spiral, if it managed to regain solvency with higher premiums, but it's *close* to a true death spiral. Take my first diagram. And suppose that administrative costs rise a small amount. We would get a sort of multiplier effect where the equilibrium premium rises a lot, and a lot of customers stop buying insurance. I *think* I could get the same sort of multiplier effect by assuming the blue MEL curve got steeper?

[Determinant]

Chris Auld

Determinant, "lemons" are not high-risk people per se, they're people who are high risk conditional on what the insurer can observe and legally contract over.

Not strictly relevant to my post. Facility Association exists to fulfil the mandate in private-auto insurance provinces that all drivers must carry insurance, and to allow all people to drive, or in other words not to exclude people from driving based on the fact that they can't get insurance.

From memory, Nick, GEICO gave up its book of business is Massachusetts and New Jersey in the 1970's since it could not secure a satisfactory rate increase from the Insurance Commissioner in those states.

[Lord]

As a buyer, one also have to consider the probability the insurer will honor the coverage, the aggravation and cost to me of getting them to do so, the loss of options to me of how to deal with loss once committed to insurance. These costs can make even personally actuarially fair insurance undesirable if one has other options, like self insurance.

[anon]

I was kind of expecting a post on adverse selection/moral hazard from a wider market equilibrium POV, as an instance of constrained-Pareto-efficiency. Perhaps the literature is a bit too recent for that. But the key intuition here is that asymmetric information causes "missing markets"; the 'lemons' (low-quality group) get the very same deal that they would face under perfect information, while other types ('peaches') tend to be shut out of the market.

What's more interesting, you can view policies for dealing with asymmetric information (such as mandates, fees etc.) quite simply as changes in property rights (endowments) that shift which markets are active, and what default outcomes agents get in the absence of market exchange. I think students in economics should be exposed to this approach in some form, as it works quite well as a sensible generalization of the 'ordinary', perfect-information case.

[anon]

Did my comment end up in the spamtrap too? This is getting annoying. (admins, feel free to delete this comment as soon as the issue is resolved...)

[Nick Rowe]

anon: yes, I found both your comments in the spam trap! (I left your second one published to try to train the spam filter to recognise you.)

Interesting perspective. New to me.

[Matt Lajoie (student)]

"That equilibrium is not efficient. It fails to maximise consumers' surplus (producers' surplus is zero by assumption). The efficient allocation is where every individual buys insurance. To see this, note that the marginal cost curve is the blue curve, and the marginal benefit curve is the red curve, and marginal benefit exceeds marginal cost for every individual. If insurance companies could price-discriminate then every individual would buy insurance. But price discrimination would require we drop the assumption of asymmetric information. If the government made insurance mandatory for all individuals, the premium would drop to where the green curve cuts the 100% line, but the lower risk individuals would be worse off as a result. (But the winners could in principle compensate the losers.)"

It seems to me we should remember that the source of the inefficiency here is asymmetric information. Although mandatory government insurance for all individuals looks more "efficient" for the moment (with winners more than able to compensate losers), it undermines potential real gains in efficiency which would be brought about by market forces (it also shifts some of the cost of this information problem onto the "peaches"). If poor information is the problem, then better information is the solution: If firms face incentives to be better price discriminators, then real inroads will be made in solving the actual problem, and REAL gains in efficiency can take place. There are degrees of asymmetry, and they aren't handed down from on high. Treating information as an exogenous variable is misleading and unrealistic; I think there are more moving parts to the practical problem than your diagrams show.
Perhaps what I have said is outside the scope of this post, but hopefully it adds some colour here.

[Nick Rowe]

Matt; fair point. Yep, any model is only ever a part of the big picture.

[anon]

Matt and Nick, building on my previous comment, it seems easy to have a solution that leaves the "peaches" better off and preserves informational incentives. Simply institute a default mandate, as before, _but then also allow agents to sell their insurance rights back to the insurer_, at individually contracted prices. Now the asymmetric info problem is reversed: insurers fear that only "peaches" will sell their insurance back, so the resale price _drops_ and non-peaches (including "lemons") are shut out of the market! But it's okay, because the default outcome now is being within the insurance pool, as opposed to being uninsured, which is where the overall efficiency gain is coming from. And, as a bonus, we allow "peaches" to uninsure if they so choose, at perfect information prices, so they don't bear any cost wrt. the first-best outcome.