Review of the Last Lecture
• began our discussion of market failures
• looked at what a market failure is
• listed four sources of market failure
• noted that health insurance is a source of market failure
• then began our discussion of why, despite health insurance causes a
market failure, there is nevertheless a demand for health insurance =>
basic reason => people are risk averse
• risk averse => declining marginal utility of wealth
• today look at the utility gain from health insurance
317_L14, Feb 6, 2008,
J. Schaafsma
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Expected Wealth and Expected
Utility: No Health Insurance
• assume the person is risk averse
• let q = probability of becoming ill, thus (1 – q) = prob not ill
• let L = financial loss if ill
• let W0 = wealth if not ill, thus wealth if ill = W0 – L
• in the absence of insurance, expected wealth is (SEE DIAGRAM):
•
•
E(W) = q(W0 – L) + (1 – q)(W0)
= W0 – qL
• in the absence of insurance, expected utility is (SEE DIAGRAM):
•
E(Utility) = qU(W0 – L) + (1 – q)U(W0)
317_L14, Feb 6, 2008,
J. Schaafsma
2
Expected Wealth and Expected
Utility: With Health Insurance
• assume insurance can be purchased at an actuarially fair premium
(premium = expected loss = qL)
• this assumption is convenient to illustrate our point but unrealistic ~>
insurance companies incur admin costs and also need a normal return ~>
in real world, premium will exceed qL (will discuss later)
• if insurance purchased, wealth = W0 - qL (same as without insurance)
• if insurance purchased, utility will be U(W0 – qL)
• N.B. Because of declining MU of Wealth:
•
U(W0 – qL) > qU(W0 – L) + (1 – q)U(W0) (SEE DIAGRAM)
317_L14, Feb 6, 2008,
J. Schaafsma
3
Utility Gain From Health
Insurance
• utility gain from health insurance is the difference between the utility
from (wealth - insurance premium), and the expected utility in the
absence of insurance (SEE DIAGRAM)
• without insurance the combination of expected utility and expected
wealth lies on the straight line joining the two extreme outcomes without
insurance (wealth and utility if ill and not ill), since the same weights are
used in computing expected wealth and expected utility [q and (1-q)].
• if MU(W) constant, no gain from insurance (see diagram)
317_L14, Feb 6, 2008,
J. Schaafsma
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Utility Gain When the Insurance
Premium > Expected Loss
• can’t purchase insurance at a premium = expected loss (qL)
• insurer needs to charge more than the expected loss to cover the cost
of running the business and to earn a return on investment. This extra
amount is caller the load factor.
• insurance premium = expected loss + load factor
• Thus wealth with insurance is (W0 – premium) < (W0 – qL)
• with the load factor included in the premium ~> utility associated with
insurance  since W 
• load factor reduces potential gain from insurance (see diagram)
317_L14, Feb 6, 2008,
J. Schaafsma
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q, L and the Utility Gain from
Health Insurance
• holding the expected loss, qL, constant the gain from insurance rises
as q  and L . (see diagram)
• if q1 < q2 and L1 > L2 and q1L1 = q2L2 ~> utility gain from insuring
against large loss with a low probability is greater than the utility gain
from insuring against a small loss with a high probability (provided the
expected losses are the same).
317_L14, Feb 6, 2008,
J. Schaafsma
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