This past Thursday, we discussed the logic behind the Rothschild-Stiglitz “separating equilibrium” model (see pp. 22-24 of the Moral Hazard and Adverse Selection lecture note) and provided a numerical illustration of its inner workings .
From our initial study of adverse selection in insurance markets (see pp. 17-21 of the Moral Hazard and Adverse Selection lecture note), we find that low-risk insureds cross-subsidize high-risk insureds when pooled premiums (based upon the average of the expected costs for both risk-types) are charged. In the “dynamic” version where there are many different risk-types (see Adverse Selection Dynamics Class Problem), this results in the so-called “insurance death spiral”. The death spiral begins with the exit of the lowest risk members of the pool, because pooling makes insurance too expensive for them and they are better off self-insuring. Their exit causes premiums for remaining pool members to increase, which motivates even more lower risk members to also exit, further shrinking the risk pool and making it even more expensive. Unchecked, this dynamic ultimately results in the failure of the insurance market.
The purpose of the static (2 risk-types) Rothschild-Stiglitz model is to show how insurance contract design can mitigate the adverse selection problem described in the previous paragraph. By offering full coverage contracts (based on the high-risk loss probability) and partial coverage (based on the low-risk probability), the high-risk and low-risk types credibly confirm whether they are high-risk or low-risk by their contract choices. Since the full coverage contract provides high-risk types with greater expected utility than the partial coverage contract, and the partial coverage contract provides low-risk types with greater expected utility than the full coverage contract, voilà – the adverse selection problem goes away because the insurer now knows who’s who!
The problem that we worked on toward the end of Thursday’s class meeting provides a numerical illustration of the Rothschild-Stiglitz model. Here is the problem description (from pg. 24 of the Moral Hazard and Adverse Selection lecture note):
Note that Policy A represents actuarially fair full coverage based on the high-risk probability, whereas Policy C represents actuarially fair partial coverage based on the high-risk probability. Without any further calculation, the Bernoulli principle implies that high-risk types will prefer Policy A over Policy C, and that Policy A and Policy C are preferred to self-insurance. Furthermore, Policy B will never be offered, since high-risk types prefer Policy B over A and the insurer would lose $19.50 ($65-$45.50) per high-risk type if it offered Policy B.
Since we are interested in determining the policy pair which maximizes (expected value of) profit, it all boils down to whether the insurer offers Policy C or D. We already know that the high-risk types prefer A over C. We need to determine whether the low-risk types prefer C or D, and whether there’s any possibility that high-risk types might defect from A to D if D were offered (note that the choice of D over A by high-risk types loses money for the insurer, since the expected cost of 30% coverage of high-risk types costs $19.50, and policy D’s premium is only $13.65). Furthermore, while we know that high-risk types prefer A to C, we don’t yet know the preference ordering by low-risk types of self-insurance, Policy C, and Policy D. Under Policy C, the expected profit per low-risk type is $39 – .6(35) = $18, but it is only $13.65-$10.50 = $3.15 under Policy D.
The following spreadsheet provides with the answers that we need (clicking on the picture below brings up the spreadsheet from which this picture is obtained; see the worksheet labeled as “RS (Class Problem)”):
The various calculations in this worksheet confirm our intuition – the profit maximizing pair is A and C. If A and C is offered, then the insurer earns expected profit of $0 on A per high-risk type (because A is purchased exclusively by high-risk types) and expected profit of $18 on C (because C is purchased exclusively by low-risk types). If Policy D is offered instead of Policy C, then high-risk types still prefer A (and yield expected profit per high-risk type of $0), whereas low-risk types prefer D (and yield expected profit per low-risk type of $3.15)