Two risk averse people are identical in every way (including wealth, W , amount of loss, L, and utility function, U(W)) except they have different probabilities of loss. Suppose they want to buy insurance from a risk neutral monopoly insurance company. Let p1 and p2 denote the probabilities of loss of each agent, respectively, with p1 < p2. The company cannot distinguish between the two agents. An insurance policy is a pair (P;I) where P is the premium paid and I is the level of coverage (amount paid out in case of loss).
(i) If the company is restricted to o§er a single full insurance policy what would it be?
I have found its solution: but I don’t understand a point in its solution.
We consider two situations.
1st Situation: we consider a full insurance only to low types (I.e. people who have high probability of loss (p2))
Let’s characterize the insurance policy
$W-P-L-I=W-P \iff L=I$
I understand how to write these three steps perfectly. But,
2nd Situation: we consider a full insurance only to both high types and low types (I.e. people who have high probability of loss (p2) and who have low probability of loss (p1))
I am stuck at this point,
- $W-P^L-L-I=W-P^H \iff L=I$
- Firm’s expected profit = $(1-p_1)P^H+(p_1)(P^H-I)+(1-p_2)P^H+(p_2)(P^H-I)$
In yellow box, in 1st and 2nd steps, why do the one side of the equation include $p^H$ and other side of the equation include $p_L$?
And in the 3rd step, why price level don’t include wealth W?
For example, if I solved the question by myself, I wrote the equations in yellow box as follows :
2. $W-P^H-L-I=W-P^H \iff L=I$
4. Firm’s expected profit = $(1-p_1)P^H+(p_1)(P^H-I)+(1-p_2)P^L+(p_2)(P^L-I)$
But what I write above are wrong. why?
In summary, I don’t understand how to write equations in the yellow box? Please do a short explanation for those.
If you have an idea, I will be really glad if you explain.