# Insurance question: need a brief explanation

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

1. $p_2U(W-P-L-I)+(1-p_2)U(W-P)=p_2U(W-L)+(1-p_2)U(W)=U_L$

2. $W-P-L-I=W-P \iff L=I$

3. $P^*=W-u^{-1}(u_L)$

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,

1. $p_1U(W-P^L-L-I)+(1-p_1)U(W-P^H)=p_1U(W-L)+(1-p_1)U(W)=u_H$
2. $W-P^L-L-I=W-P^H \iff L=I$
3. $P^*=u^{-1}(u_H)$
4. 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 :

1. $p_1U(W-P^H-L-I)+(1-p_1)U(W-P^H)=p_1U(W-L)+(1-p_1)U(W)=u_H$

2. $W-P^H-L-I=W-P^H \iff L=I$

3. $P^*=W-u^{-1}(u_H)$

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.

• Could you explain how you get the equations in the 1st situation, the one you understand perfectly, in detail? It seems there is very little difference between it and the 2nd situation, so your confusion is somewhat unclear. – Giskard Jul 11 '18 at 22:15
• Sorry typo, I edited. They are the same. – Enjoyecon Jul 11 '18 at 22:16
• In the first one, full insurance policy is offered only to low type. (Note if it is offered to high types, then low types will deviate.). 1. With prob of loss p2, the low type agent’s utility level when loss occurs plus with probability of no loss, the agent’s utility when she doesn’t experience any loss under insurance policy should be equal to the her expected utility under no insurance. This step should be equal in order for the poilicy to be optimal. – Enjoyecon Jul 11 '18 at 22:21
• In the second part, we show that full insurance I.e. loss = coverage. – Enjoyecon Jul 11 '18 at 22:21
• I can show mathematically how two derive price level in first case( if you want). But I cannot understand in the yellow box, why does one utility include $p^H$ and another utility includes $p^L$ and how to derive price level in yellow box? @denesp – Enjoyecon Jul 11 '18 at 22:23