Replacing a car

Problem definition

At the beginning of each year, a car is in good, regular or bad state:

• There is a 90% probability that a good car will be in a good state at the beginning of next year

• There is a 5% probability that it will be in a regular state

• and a 5 % probability that it will be in bad state.

If a car is in a regular state, then:

• There is a 70 % probability that the car is in again in regular state at the beginning of next year,

• and a 30 % probability that it will be in a bad state.

A bad car has no sale value and must be immediately replaced with a good one.

It costs €12,000 to buy a good car, but a regular one can be found for €5,000. Maintenance costs are €1,000/year for a good car and €2,000 for a regular car, and must be paid upfront at the beginning of the year.

Must I replace my car as soon as it is in a regular state, or must I wait until it is in a bad state?