Automobile insurance companies insure both safe and unsafe drivers. Ideally they would like to be able to charge different premiums to each to reflect the cost differences in insuring both types of drivers

What type of economic problem is this and how might insurance companies deal with it?

This is a problem of imperfect information. There are a variety of ways in which insurance companies can deal with this issue. First, insurance company can examine the motorist's driving record to determine if there has been a history of moving traffic violations. The age of the driver might also be a factor as insurance companies are well aware that lack of experience is a major cause of accidents. Lastly it can require that drivers pay a deductible when there is an accident. Insurance companies could charge higher premiums to motorists who choose lower deductibles and vice versa. These are methods by which the insurance company is attempting to push part of the cost burden back to the motorist.