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.)
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.)