Question 4: (Demand for Insurance) Suppose you are a risk-averse expected utilit

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Question 4: (Demand for Insurance) Suppose you are a risk-averse expected utility maximizer with utility function u). 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 $100 $500 $1500 $10,000 Probability 20% 8% 2% 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 pp". 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. (d) If the coinsurance rate goes up to 15%, what happens to your p"? Briefly explain your answer

Explanation / Answer

a.) Without Insurance =>

20000 - E(medical expense)

= 20000 - (100*20% + 500*8% + 1500*2% + 10000*1%)

= $19810

With Insurance,

=> 20000 - 10%(E(medical expense)) - p

= 20000 - 10%(100*20% + 500*8% + 1500*2% + 10000*1%) - p

= 19990 - p

b.) To buy insurance, wealth left with insurance > wealth left without insurance

=> (19990 - p) > 19810

=> p < 180

therefore, p* = $180

c.) Risk-Neutral => p* = $180

RIsk-Averse => p* > 180 (Insurance preferred to lessen the risk, willing to pay more money)

As risk-aversion increases, p* increases

d.) If co-insurance goes upto 15%, we have to pay more in case when we buy insurance, therefore p* will go down.

Now, with insurance wealth left,

=> 20000 - 15%(E(Medical Expenses)) - p > 19810

=> 20000 - 15%(100*20% + 500*8% + 1500*2% + 10000*1%) - p > 19810

=> 19985 - p > 19810

=> p < 175

=> p* = 175$ which is lesser than the earlier case.

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