Suppose you are a risk-averse expected utility maximizer with utility function u(x). You have initial wealth $20,000, but before you consume it, you are subject to the following he... Suppose you are a risk-averse expected utility maximizer with utility function u(x). You have initial wealth $20,000, but before you consume it, you are subject to the following health risk (these events are mutually exclusive): Required Medical Payment Probability $100 20% $500 8% $1500 2% $10,000 1% Now suppose that an insurance agent offers to sell you health insurance with 10% coinsurance — that is, for any medical payment that you require, you must pay for 10% of the payment and the insurance company pays the rest. The price of this insurance is p. (a) If you do not buy the insurance, what lottery do you face? If you buy the insurance, what lottery do you face? (b) Let p∗ denote your willingness to pay for the insurance — so that you prefer to buy the insurance if p < p∗ and you prefer not to buy the insurance if p > p∗. Provide an equation from which you could derive p∗. (c) If you are risk-neutral, what can we say about your p∗? If you are risk-averse, what can we say about your p∗? As you become more risk averse, what happens to your p∗? Briefly explain your answers. Read more

Steps to Solve

1. Define the Lottery Without Insurance

First, identify the possible outcomes and their probabilities without insurance. If a person might face a health cost of $C$ with a probability $p$, the utility function could be represented as:
$$ U(W - pC) $$
where $W$ is the initial wealth.

2. Define the Lottery With Insurance

Next, define the situation with insurance. If the insurance premium is $P$, the utility becomes:
$$ U(W - P) $$
The outcome would not depend on health costs since the insurance covers it.

3. Set Up Expected Utility Equations

Now, set the expected utility equations for both scenarios. For no insurance:
$$ EU_{\text{No Insurance}} = pU(W - C) + (1 - p)U(W) $$
And for insurance:
$$ EU_{\text{Insurance}} = U(W - P) $$

4. Determine Willingness to Pay

To find the willingness to pay for insurance, set the expected utilities equal:
$$ EU_{\text{No Insurance}} = EU_{\text{Insurance}} $$
This leads to:
$$ pU(W - C) + (1 - p)U(W) = U(W - P) $$

5. Solve for P

Rearranging the above equation will allow you to express $P$ in terms of the other variables. This will give you an equation to find the willingness to pay for the insurance.

6. Analyze Risk Attitudes

Consider how different risk attitudes (risk-averse, risk-neutral, risk-seeking) affect the value of $P$. Risk-averse individuals would be willing to pay more for insurance compared to risk-neutral or risk-seeking ones due to the declining marginal utility of wealth.