Question

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
$6K
$14K
Maximal price the customer is
ready to pay for insurance
$3K
$7K
$15K

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 $7K (in which case only customers with medium to high risk will buy the policy), or $15K (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.

Answer 1

A. Average costs= Total cost/number of customers.

Number of customers is given as 170000.

Total costs= 100000*2000+60000*6000+10000*14000

Total costs= 700,000,000

Average cost= 700,000,000/170000=~4118

B. No. Low risk profile customers will not buy since the maximum they are willing to pay is $2000.

C. Let us look at all 3 scenarios.

If the company charges 3k, then all customers will accept the policy. The company will earn

3000*170000=510,000,000.

As discovered in part A, the cost incurred by the company would be 700,000,000. Hence,

Profit in this case=510,000,000-700,000,000=-190,000,000.

There is a net loss if the company charges $3k.

Case 2- The company charges 7k.

In this case, low risk customers will not avail.

Total revenue=70000*7000=490,000,000

Total cost= 60000*6000+10000*14000= 500,000,000.

Net profit=490,000,000-5,000,000=-10,000,000.

There is net loss if the company charges $7k.

Case 3- If the company charges 15k

In this case only high risk customers would avail.

Total revenue= 10000*15000=150,000,000

Total cost= 10000*14000=140,000,000.

Net profit=150,000,000-140,000,000=10,000,000

Since there is a profit only in case the company charges $15k, the company will charge $15k.

d. In this case, the company will simply sell the policy at the maximum price each customer is willing to buy it at. So it will sell it at 3k to low risk customers, at 7k to medium risk customers and at 15k to high risk customers. The cost will remain the same as calculated in part A, which is 700 million, but the revenue will change.

New revenue=100000*3000+60000*7000+10000*15000= 870,000,000.

Hence,

Net profit=870m-700m=170m

A profit of 170m is far higher than the max profit of 10m in part C.