In this simple insurance model, a company has a monopoly over a small market. There are 100K potential customers with a low risk profile, 60K potential customers with a medium risk profile, and 10K potential customers with a high risk profile. A person’s risk profile is important as it determines how much insurance is worth to the customer and how much money the customer will cost on average to the insurance company. The following table summarizes the estimates put together by the company: Low risk profile Medium risk profile High risk profile Number of potential customers 100,000 60,000 10,000 Expected expense per customer $2K $7K $15K Maximal price the customer is ready to pay for insurance $3K $8K $16K Remark: Explaining where these numbers come from would require a subtler model that describes the risk covered by the insurance policy. While there is no need to do this for the purpose of this exercise, notice though how the maximal price a customer is ready to pay is always larger than the expected expense the insurance company would incur for that customer. This is the case, for instance, if potential customers are risk averse while the insurance company is risk neutral.
In this simple insurance model, a company has a
|
|
Low risk profile |
Medium risk profile |
High risk profile |
|
Number of potential customers |
100,000 |
60,000 |
10,000 |
|
Expected expense per customer |
$2K |
$7K |
$15K |
|
Maximal price the customer is |
$3K |
$8K |
$16K |
Remark: Explaining where these numbers come from would require a subtler model that describes the risk covered by the insurance policy. While there is no need to do this for the purpose of this exercise, notice though how the maximal price a customer is ready to pay is always larger than the expected expense the insurance company would incur for that customer. This is the case, for instance, if potential customers are risk averse while the insurance company is risk neutral.
QUESTIONS:
(a) What is the average cost per customer if the insurance company insures all 170K potential customers?
(b) The number computed in (a) is thus the minimal price the company would need to charge to make it profitable to serve everyone. Assume here that customers know their risk profile. Would all potential customers want to buy insurance at that price?
(c) Suppose the insurance company chooses the price at which it sells its policy. Consider a classic case of asymmetric information: customers know their risk profile, but the insurance company cannot identify the risk profile of its potential customers. By deciding to sell at a price $p, all customers with a maximal price larger or equal to $p will buy the policy (and the firm must incur the expected expense associated to its customers, that is, it cannot renege on the terms of its policy). At which price will the insurance company sell its policies (assuming it aims to maximize profit)? What is the profit it realizes?
Hint: The company will always charge the maximal price customers of some risk profile are ready to pay. So it will charge either $3K (in which case customers of all risk profiles will buy the policy), or $8K (in which case only customers with medium to high risk will buy the policy), or $16K (in which case only high risk customers will buy the policy). What scenario gives the best profit?
Remark: This question illustrates well the concept of adverse selection. Notice how customers who are ready to pay more for the policy are also more costly to the insurance company.
(d) Suppose now the company can identify each potential customer’s risk profile (e.g. by doing a thorough physical exam in case of some medical insurance). To maximize profit, at what price will it sell its policy to low risk customers, at what price will it sell its policy to medium risk customers, and at what price will it sell its policy to high risk customers? What is the total profit in this case, and how does it compare to profit in (c)? This should illustrate the substantial loss in profit that asymmetric information can generate.
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