Each extra dollar adds less
The curve shows the total utility (satisfaction) a person gets from each level of wealth they are sure to have. Wealth means everything they own, in dollars. At $50,000 their utility is 140. At $100,000 it is 200.
Doubling wealth does not double utility. The first $50,000 buys necessities and matters a lot; the second $50,000 adds only 60 more.
Diminishing marginal utility of wealth
The amber line is the slope of the curve: the extra utility from the next $1,000, called the marginal utility of wealth. It gets flatter as wealth rises, so the curve is bowed (concave). This shape is what makes people dislike risk.
Try this: move wealth from $50,000 to $60,000, then from $90,000 to $100,000. Both add $10,000. Which adds more utility? You can also drag the point along the curve.
Facing a risk
Now the person has $100,000 but may fall ill. Illness brings $50,000 of medical bills, leaving $50,000. Without insurance, the year ends at B (well) or at A (ill).
Expected wealth = (1 − p) × $100,000 + p × $50,000
=
Expected utility = (1 − p) × 200 + p × 140
=
Here p is the chance of illness. Expected utility lies on the red line from A to B, at point C above expected wealth. The likelier illness is, the closer C slides to A.
An average, not an outcome
No one actually ends the year at C. Each person lands at A or at B; C is the average over many such years or people.
Try this: set the chance of illness to 0.5. Where does C land, and what is expected utility?
A sure thing beats the gamble
An insurer offers full coverage: pay a premium, and if illness strikes it pays the $50,000 of bills. Suppose the premium equals the expected payout, p × $50,000. This is an actuarially fair premium: the insurer’s expected payouts just equal the premiums it takes in.
Well: $100,000 − premium. Ill: $50,000 + $50,000 − premium.
Either way: , for sure.
Same average wealth, more utility
Point D is on the curve directly above C. The person has the same expected wealth either way, but a sure amount gives more utility than a gamble with that average. The green bar running straight up from C to D is the utility gained by removing the risk; its size is printed beside the utility scale.
Try this: is there any chance of illness where D falls below C? What about the curve’s shape guarantees your answer?
The most they would pay
Real insurers also cover administration and other costs, so premiums run above the fair level. How high can a premium go before the person would rather stay uninsured?
Follow the amber line left from C to the curve, at point F. A sure $80,000 gives the same utility, 194, as the gamble. So the person would pay up to $100,000 − $80,000 = $20,000 for full coverage.
The segment FC measures risk aversion
Of that $20,000, $5,000 is the expected loss (the fair premium). The other $15,000, the length of the highlighted segment FC, is the risk premium: what they would pay just to be rid of the risk. The more bowed the curve, the longer FC.
Why this range? Below the fair $5,000 the insurer would expect to pay out more than it collects, so no insurer offers it; that would put I to the right of D. Above $50,000 the premium would exceed the loss itself. At $50,000, I reaches A: paying the whole loss for sure leaves the person as badly off as being ill.
The bracket along the bottom marks the premium, and point I shows where it leaves them, insured. The bar at I measures the change in utility against the gamble’s level (194):
- Green, above: insuring gains utility, so they buy.
- Red, below: insuring loses utility, so they stay uninsured.
- At $20,000 I lands on F: no gain or loss, so they are indifferent.
- At $5,000 (the fair premium) I is at D, the largest gain.
The difference between the most they would pay and the premium is their consumer surplus.
Try this: drag I along the curve. At what premium does it turn red?
Explore
Change the chance of illness and the premium together. The bar along the bottom of the graph splits the most they would pay into the fair premium (gray, the expected loss) and the risk premium (amber, the extra paid only to remove risk).
What the graph shows
- People pay more than the fair premium because marginal utility of wealth diminishes: they are risk-averse, so removing risk is worth something to them.
- A premium above the fair level leaves them with less expected wealth than staying uninsured. They still buy, because removing the risk raises their utility.
- The higher the chance of illness, the more they would pay in total, because the expected loss grows.
Watch the amber part. The risk premium does not keep rising with the chance of illness. It peaks near p = 0.3 here and shrinks toward zero as illness becomes almost certain: a near-certain loss is an expected cost, not much of a risk.