Final answer:
In a perfectly competitive insurance market where insurers cannot distinguish between reckless and safe drivers, the insurer would charge a single premium by calculating the weighted expected losses for each type of driver using the proportion of reckless drivers (β). The problem described shows adverse selection, where high-risk individuals benefit more from insurance, potentially leading to market failure.
Step-by-step explanation:
The scenario outlined is a classic example of adverse selection, a term used in economics and insurance to describe a situation where an insurer cannot distinguish between high-risk and low-risk customers, leading to the high-risk customers benefiting disproportionately. In the scenario, reckless drivers represent a high risk with an expected loss of $5,000 to the insurance company, while safe drivers represent a low risk with an expected loss of $1,000. The proportion of reckless drivers in the total population is given by the variable β.
If we assume a perfectly competitive insurance market, the insurer would set a single premium that would cover the expected losses weighted by the proportion of each type of driver in the population. To calculate this, the insurer would use the formula:
Premium = β * (Expected loss for reckless drivers) + (1 - β) * (Expected loss for safe drivers)
Assuming the market is risk-neutral, the insurer would charge a premium equal to:
- β * $5,000 + (1 - β) * $1,000
However, due to adverse selection, if the premium is set too high, safe drivers may opt out of purchasing insurance, leaving only the high risk drivers in the pool, which could lead to losses for the insurer as described in the referenced information.