Failure of the Separating Equilibrium:

We are now  in a position  to examine the policies that constitute a separating equilibrium above. See that  in Figure, both types, H and L, would buy the policy D because  of  its  location above each of  their indifference curves when purchasing policies AL  and C. However, D is a pooling policy. As we have already seen  above, it  cam01  exist  in  equilibrium.  In  the following therefore we will examine the circumstances when D  can be offered by  an insurance firm.  

It  is easy  to  see that  the insurance company would  offer D when it  is more profitable than  that of  the  pair  (A_L, C), which yields zero profits. The profitability of D in turn is determined by  the share of high-risk claimants in the population. Let us define such a  share as  A. From  the figure,  it  can be seen that the slope of the fair odds line for pooling policies depends on A. So,

i)  when the value ofA  is larger, pooling odds line lies closer to  the H fair odds line;

ii)  when the value of A  is smaller, the pooling odds line lies closer to the L fair odds line.

Construct two  different cases  λ+ (greater  share of high risk population) and λ-  (greater share of low risk population) such that these give the pooling odds lines. When you  consider  λ+(i.e., population  is  mostly  high  risk type),  a pooling policy D will lead to breaking up of separating equilibrium. Since D lies above  the fair  odds line,  it  becomes unprofitable and hence  cannot  be offered.  By  contrast, consider  λ-  (i.e.,  population  is mostly low  risk type).

The pooling policy D is profitable here  since it  lies below the fair odds line. But it breaks the separating equilibrium.