Joe is currently unemployed and without health insurance coverage. He derives utility (U) from his interest income on his savings (Y) according to the following function: U = 5(Y1/2)

Joe presently makes about $40,000 of interest income per year. He realizes that there is about a 5 percent probability that he may suffer a heart attack. The cost of treatment will be about $20,000 if a heart attack occurs.

Calculate Joe’s expected utility level without any health insurance coverage.

Calculate Joe’s expected income without any insurance coverage

Suppose Joe must pay a premium of $1,500 for health insurance coverage with ACME insurance. Would he buy the health insurance? Why or why not?

Suppose now that the government passes a law that allows all people—not just the self-employed or employed—to have their entire insurance premium exempted from taxes. Joe is in the 33 percent tax bracket. Would he buy the health insurance at a premium cost of $1,500? Why or why not? What implications can be drawn from the analysis?

1. Joe's expected utility without any insurance coverage is $985.36

2. Joe's expected Income without any insurance is $39,000

3. Joe will not buy insurance; if he buys insurance, it'll lower his income (and utility) to below the level he can expect to obtain without purchasing the insurance.

4. Yes, he will (See Explanation Below)

Explanation:

Given

Function,U= 5(Y^½) where Y = Savings

Let P = Chances of having a heart attack = 5% = 0.05

Let Q = Chances of not having a heart attack = 1 - 5% = 1 - 0.05 = 0.95

Let C = Cost of Treatment = $20,000

Let A = Income per year = $40,000

1. Expected utility without any insurance coverage is calculated as follows:

Expected Utility = 5PC^½ + 5QA½

Substitute respective values in the above equation

Expected Utility = 5 * 0.05 * √20,000 + 5 * 0.95 * √40,000

Expected Utility = 985.3553390593273

Expected Utility = 985.36 Approximated.

Hence, Joe's expected utility without any insurance coverage is $985.36

2. Expected income without any insurance coverage is calculated as follows

Expected Income = QA + P(A-C)

Expected Income = 0.95 * $40,000 + 0.05 * ($40,000 - $20,000)

Expected Income = $39,000

Hence, Joe's expected Income without any insurance is $39,000

3. First, we'll calculate his expected Loss.

Expected Loss is calculated as = ∆Income

∆Income = Difference in Income if he has insurance and if he doesn't

Income if he has insurance = $40,000

Income if he doesn't = $39,000

Expected Loss = $40,000 - $39,000

Expected Loss = $10,000

He has an expected loss of $1,000.

U40,000= $1,000, compared to U39,000 = $987.42, and U38,500= $981.07.

Joe will not buy insurance

If he buys insurance, it'll lower his income (and utility) to below the level he can expect to obtain without purchasing the insurance.

4.

Assume that Joe is taxed at 33% rate, the following analysis applies

Annual tax = 33% of $39,000 = $12,870

He's left with $39,00 - $12,870 = $26,130

Utility = $808.24

If premium = $1,500, the following analysis applies;

Income = $40,000

Taxable Income = $40,000 - $1,500 = $38,500

Annual tax = 33% of $38,500 = $12,705

He's left with $38,500 - $12,705 = $25,795

Utility = $826.06

If he chooses tax free insurance, the following analysis applies.

Utility is greater

If premium is tax exempt, he'll pay insurance

This means that incentives can apply for people with good investments.

As it is, currently the US is making attempts to do with fines for failure to carry health insurance under ACA.