How Insurance Works

The Demand for Insurance


Okay, we want to now reinterpret the same
model that we looked at as a situation where the consumer is going to perhaps experience
a loss with a certain amount of probability. So the good state where there’s no loss the
income is 2250. The loss is 2000 and in the bad state they have only 250. What insurance
is supposed to do is transfer income from the no loss state to the loss state. So we’re
going to move down and to the right to more equalize income. Insurance shifts income over
states, like if you will remember, borrowing and saving shifted income over time. Before I get to the analysis in this picture
let me quickly show that in the spreadsheet by scrolling to the right here, and then I’ll
scroll down, there is a written discussion of insurance and the insurance model. So you
should read through that on your own. I am not going to do that here. I’m going to focus
on the graph and the mechanics of the graph. The picture is supposed to show that this
looks like choice with indifference curves and budget line. This gray line is the only
thing that’s new here. And that is the insurance line. You can see this little dashed indifference
curve is tangent to the insurance line. It happens to be lower than the original indifference
curve. So this is a case where the person wouldn’t buy any insurance and would stay
right at the original point because the insurance is too expensive. You have some additional controls to what
we had before. So in addition to the risk of aversion control there’s a control for
the fixed load of the insurance. The fixed load is like a fixed cost. It doesn’t vary
with how much insurance you buy. And then there is a variable load for the insurance.
That affects the price per dollar of coverage. And right now the coverage per dollar is set
to be at the same price as the probability of loss. This is called fair coverage. When
v equals p that is fair coverage. Otherwise the coverage is biased. So let’s take a quick look here. I’m going
to be lowering F and look what happens to the insurance line as I lower F. I’m going
to lower it a lot and see what happens. So let’s lower it quite a bit. And lower it still
more, still more,
still more. Okay, we’ve got it low enough now where buying insurance is actually a better
thing to do. And let’s make it even higher. Buying insurance is pushing this consumer
to a higher indifference curve. Just to make the picture clear I’m going to
increase the Arrow-Pratt measure of risk aversion so that we can make the insurance graph more
clear. So there’s the insurance line. The consumer ends up choosing certainty. Under
fair insurance that is the result. The marginal rate of substitution actually equals the odds
at the certainty point. And so does the fair insurance,…. absolute value of the slope
of the fair insurance line also equals the odds. So the tangency point happens at certainty. If I make insurance unfair and let’s now increase
v so that we actually make v much bigger than p, then the consumer buys less than full coverage.
If I make v much less than p, and I’m going to do that now, they buy more than full coverage.
And when v equals p they buy exactly full coverage. That’s the result to get from this
section.


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