Answer:
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.
