Alex has $20,000, and his utility function is U = W0.5, where W is his wealth. H

Question

Alex has $20,000, and his utility function is U = W0.5, where W is his wealth. His probability of having a heart attack is p and the treatment of a heart attack costs $10,000. A heart attack must be treated regardless whether an individual has insurance or not. Write Alex's expected utility, UE, as a function of p when he does not have any health insurance. Find a corresponding insured wealth level, Wt, that gives Alex the same utility level as UE. Write his expected wealth, Wg, as a function of p when he does not have any health insurance.

Explanation / Answer

U = W

Probability of heart attach = p

wealth = 20000
when heart attack occurs without insurance, wealth = 20000-10000 = 10000

U = W = 20000 = 141.42

(a)

When alex has no insurance:
U = p*(U when heart attack occurs) + (1-p)*(U when heart attack does not occur)
= p*((20000-10000)) + (1-p)*(20000)
=100p + (1-p)*141.42
= 141.42 - 41.42p

UE = 141.42 - 41.42p

(b)

Insured wealth level = WI
U = Wi
this is equal to UE

Wi = 141.42 - 41.42p
Wi = (141.42 - 41.42p)2

This is the corresponding insured wealth level that gives the same utility as UE
When insured, the wealth level is independent of the heart attack (p)

(c)

When he does not have insurance

expected wealth
WE = p * (wealth when heart attach occurs) + (1-p)*(wealth when heart attach does not occur)
= p * (20000) + (1-p) * (20000-10000)
= 20000p + 10000 - 10000p
= 10000(1+p)

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