In their pathbreaking paper, Rothschild and Stiglitz define the RS Equilibrium as a set of contracts such that, i) each firm breaks even ii) there exists no other contracts making non-negative profits when offered.
And they show that no pooling equilibrium exists and there may only be a separating equilibrium.
In this separating equilibrium, high risks get full coverage but low risks only partial coverage with premia equal to loss probabilities respectively.
What would you think about another separating contract bundle, in which high risks are cross-subsidised by low risks, as shown in the graph with red insurance lines and green indifference curves (G, H)?
When the new bundle is offered, high risks would prefer it to E and low risks to F (see the indifference curves). And high risks are indifferent between G and H, as in the usual separating equilibrium. Of course, this new bundle wouldn’t make an equilibrium as the yellow shaded area attracts low risks only and it might be the case similar to pooling.
However, wouldn’t the possibility of such bundles lead the usual separating contract to an instability? The new bundle making a non-negative profit seems to violate the second condition of Rothschild-Stiglitz equilibrium.
- Utility function: natural log
- Initial wealth=100
- Loss in case of accident=90
- Loss probability of low risks=0.1
- Loss probability of high risks=0.9
1) Incentive Compatibility Constraint for Original RS Eq.
where x is the partial coverage amount that low risks receive in original RS eq. Matlab calculates;
- Utility(Low risks)= 4.424718438632044
2) Low risks subsidising high risks
Add 0.08 to the premium of low risks and subtract the same amount from high risks. (P.S: This is just an example, I also have the number maximising the utility)
- new coverage=15.424194558495868
- new utility(low risks)=4.452073664660258
- utility of high risks=3.265759410767051 (same for full coverage at higher premium and partial coverage at lower premium)
P.S: There is no need for computing the utility of high risks because they obviously reach to a higher utility level by still receiving a full coverage with a cheaper premium.
Finally, we see that new bundle beats the original RS bundle, doesn't it?
Below are the Matlab codes I made use of:
y=0.08 %subsidy amount
ICC = @(x) log(100-(0.9-y)*90) - 0.9*log(100-90-(0.1+y)*x+x) - 0.1*log(100-(0.1+y)*x); %Incentive Compatibility Constraint
z = fzero(ICC,50); %coverage amount of low risks WRT ICC
u=0.1*log(100-90-z*y+z)+0.9*log(100-z*y) %utility of low risks
Edit: I also might have a general algebraic proof, but I need to verify it and will share it here afterwards.