The benefit and cost of coverage
A person has $100,000 and a 0.10 chance of an illness that costs $50,000. They can buy any amount of coverage, q, at a premium of 20 cents per dollar of coverage: $1,000 for a $5,000 policy.
Coverage helps only when they are ill: each dollar of coverage pays $1 but costs 20 cents, a net gain of 80 cents. When they stay well, the premium is a pure cost.
The utility worksheet adds up the graph: the area under MB is the expected utility coverage brings when ill, the area under MC is what its premiums cost when well, and the green area between them is the net gain.
Weighing the next dollar of coverage
Marginal benefit (MB, blue): the expected utility the next dollar of coverage adds when ill. Marginal cost (MC, red): the expected utility its premium takes away when well. Both are measured in utility, so the graph’s vertical scale is an index.
Try this: slide coverage from $5,000 to $10,000. Which way do A and A′ move, and why?
The best amount of coverage
At $5,000 of coverage, the marginal benefit (A) is far above the marginal cost (A′): another dollar of coverage adds more than it costs, so the person buys more. At $10,000 (B and B′) the gap is smaller.
Buy until marginal benefit equals marginal cost
The gap closes at point X, where MB = MC. That coverage, q*, is the best amount: about $30,000, less than the $50,000 loss. Past X, another dollar costs more than it adds.
Watch the expected column as you move the coverage line. The net gain (green area) grows while each extra $1,000 adds more than it costs, peaks at X, and shrinks once the red area appears: buying past X gives back utility.
Try this: drag the coverage line to the right of X. Which curve is now higher, and what does that tell the person to do?
A higher premium
Raise the premium from 20 to 25 cents per dollar of coverage: $1,250 for the $5,000 policy. Each dollar of coverage now nets only 75 cents when ill, so MB shifts down to MB2. Each dollar also costs more when well, so MC shifts up to MC2.
The new optimum Y is at q**, about $23,000. People react to higher premiums by buying less coverage: the demand for insurance slopes down.
Try this: choose the fair 10% premium. How much coverage does the person buy now?
A bigger expected loss
Back at a 20% premium, suppose illness would cost $75,000 instead of $50,000. Wealth when well does not change, so MC stays put. But wealth when ill is now much lower ($29,000 with a $5,000 policy), where each extra dollar matters more: MB shifts up to MB2.
The new optimum Z, at q***, is higher, about $35,000: a bigger expected loss raises the demand for coverage.
Try this: raise the loss step by step from $50,000. Which curve moves, which stays put, and why?
More wealth
Now suppose the person starts with $125,000 instead of $100,000, with the same $50,000 loss and 20% premium. Two things change at once:
- The coverage pays off at a higher wealth, where it adds less utility: MB shifts down.
- The premium comes out of a higher wealth, where it costs less utility: MC shifts down too.
The new optimum W can be above or below q*: it depends on which shift is larger. Compare three people:
Try this: switch between the three people. Why can richer people buy more coverage, less, or the same?