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Suppose we have three individuals, persons A, B, and C. All three individuals face some probability of contracting an illness that would result in…

Suppose we have three individuals, persons A, B, and C. All three individuals face some probability of contracting an illness that would result in 10,000 in medical expenses. Each individual also has a maximum amount that they would be willing to pay for full insurance, based on their own levels of risk aversion. This information is summarized in the following table: 

 Probability of illness Cost of illness Max willingness to pay for full insurance

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Person A 20% 10000 3000

Person B 40% 10000 5500

Person C 60% 10000 8000 

The insurance company knows that there is one of each type in the market but cannot observe type. It wants to offer one full insurance contract on which it breaks even (does not make a profit) and insures the most people possible. What premium will the insurance company charge and who will be insured / uninsured?  

(b) Explain intuitively why the outcome in (a) results in efficiency loss.  

(c) Now suppose that the government mandates that all individuals purchase insurance and imposes such high penalties that all types do choose to purchase insurance. Again, the insurance company wants to offer one full insurance contract on which it breaks even (does not make a profit) and insures the most people possible. What premium will the insurance company charge and who will be insured / uninsured?  

(d) Now suppose instead that the government mandates that all individuals purchase insurance and charges a penalty of $500 to individuals that fail to comply with the mandate. What will happen in the market? Explain. 

2. Adverse Selection II 

Suppose that there are two individuals in the economy. Both individuals face some risk of an illness that would cost $40,000 in medical bills. The Type H individual faces a 30% chance of getting the illness. The Type S individual faces a 50% chance of getting the illness. Both individuals have income of $100,000 if the illness does not occur. Both individuals have the following utility function: income U  . 

(a) What is the expected utility of the Type H individual if she does not have insurance against this illness? Please give a numerical answer and show your calculations. 

(b) Suppose that the insurance company can observe each individual’s type and charges the Type H person a premium equal to her expected cost. If the individual purchases insurance, the insurance company will pay all medical bills if the illness occurs. What is the expected utility of a Type H person if she purchases insurance? Remember, the premium is paid in both states of the world. Please give a numerical answer and show your calculations. 

(c) Suppose your answer to (a) were 280. What is the largest premium the insurance company can charge the Type H person such that she is still willing to purchase insurance? Please give a numerical answer and show your calculations. 

(d) Now the insurance company cannot observe the type, but it knows that there is one Type H and one Type S person in the population. Suppose your answer to (c) were 15,000. Will the insurance company be able to offer a contract that allows the insurance company to break even and under which both individuals are insured? Explain.

 

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