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In this video we want to discuss moral hazard as it applies to insurance. We discussed moral
hazard earlier in the course where we talked about, for example, when somebody was buying
a house whether the realtor experienced moral hazard in helping the person buy or sell a
house for that matter. Before I get to the model, which is over here, let me show that
on this particular spreadsheet there is a rather extensive discussion and I'm not going
to go through this discussion at all but I do encourage you to read it immediately after
watching the video. It's not that long and it explains real-world insurance and some
related issues.
So here when we do the model we're going to do it algebraically only because graphing
is a little bit hard and let me explain why. In this example the consumer can affect the
probability of loss. If the consumer takes precaution they can lower the probability
of loss to q where q is less than p, the probability of loss when they don't take precaution. Because
the indifference curves themselves depend on the probability of loss, it's going to
be hard to keep track of preference when you have these two different probabilities floating
around. So we're only going to do this algebraically and we are not going to look at this analytically...excuse
me, graphically.
When they take self protection there is an effort cost, e. And we are going to assume
that the consumer is effective in taking self protection, so if you look at the expected
loss with self protection which is qL, plus the effort cost that is supposed to be less
than the expected loss without self protection. So self protection is productive here and
from society's point of view it would be good if the consumer took that.
So let's scroll down a little bit here. I'm not going to prove this algebraically but
if the original probability is not that large, it p is small enough, then the expected utility
of the lottery where W again is the initial wealth, L is the loss, so this is the probability
times the utility when you are taking precaution and you face a loss. This is probability of
the utility when you don't face the loss. That expected utility is actually bigger than
the expected utility if you don't take precaution. So this consumer has incentive to take precaution
in the absence of insurance.
And now let's scroll down a bit and what we will see here is that when they have insurance
they pay the premium S, so you subtract S in both states, but they have the coverage
I so I reduces the impact of the loss. And what becomes quite clear is that when I equals
L, that is when there is full coverage, in fact then the payoff is independent of the
state and it doesn't matter whether it's q or p. They get the same payoff. So why incur
the effort cost, e, if you have full coverage?
So with full coverage there is no incentive to take precaution. With no coverage there
is incentive to take precaution. And then you can reason that there's someplace in between
which is the maximum amount of coverage that will induce precaution. And the insurance
company will not offer more than that maximum amount. The difference between that maximum
amount and full coverge is the deductible. So this is an explanation based on moral hazard
about why insurance policies have deductibles.