Single Company Insuring Single Object

In this section the optimal insurance is regarded as a problem of insurance company. We start from the simplest case of single company insuring a single object.

The expected utility of this object

$$U(x,a)=\sum_{y} u(y) p(y),$$ (17)

where $p(y)$ is a probability of wealth $y$,
$u(y)$ is an utility function of the wealth $y$.
Here $a$ is the rate of insurance charge.

Suppose that
$y= c (x)$
and

$$c(x)= \left\{ \begin{array}{ll} z- a x & \mbox{if $\delta=1$}\\ (1-a) x & \mbox{if $\delta=0$} \end{array}. \right.$$ (18)

Here $z$ is the market value of the object,
the product $a x$ denotes insurance charge of the object.
$x \le z$ is insurance policy of the object.
$\delta=1$, if the object survives,
$\delta=0$, if not.
$p=P\{\delta=1\}$ is a survival probability of the object.

The expected utility of the insurance company

$$V(x,a)=\sum_{y} v(y) p(y),$$ (19)

where $p(y)$ is a probability of profit $y$,
$v(y)$ is an utility function of the profit $y$.
Suppose that
$y= d (x)$
and

$$d(x)= \left\{ \begin{array}{ll} a x, & \mbox{if $\delta=1$} \\ -(1-a) x & \mbox{if $\delta=0$} \end{array}. \right.$$ (20)

Here insurance policy $x$ is defined by the owner of object which maximizes his utility (15) depending on the rate of insurance charge $a$. The equilibrium between interests of the company and the customer is achieved when both insurance policy $x$ and insurance charge $a$ satisfies Nash conditions.