I use the British pounds symbol instead of dollars because $ conflicts with Mathjax.
Source: p 296, The Legal Analyst, Ward Farnsworth
"... one time in a thousand we do lose the film; if you’re willing to pay an extra ten dollars, I can have it delivered by armored car and guarantee that it won’t be lost.” ... In that case we have another way to think of the value she put on the film. If we consider the £10 a premium for a kind of insurance, we can reason backwards to the value that the owner of the film put on it for these purposes. You wouldn’t spend £10 to insure against a 1/1,000 chance of a misfortune that would cost you only £10 or £100—at least not if you were being economically rational. It wouldn’t be worth it. In the long run you would end up paying more in insurance premiums than you ever would collect when the dreaded event occurs. Imagine paying £10 a day for 1,000 days, then collecting £100 on the day when the bad thing happens: it would be a bad deal. But if the loss were worth £1 million to you, the £10 payment to avoid the 1/1,000 chance of it would be a bargain. So to return to our case, if we assume £10 is the most you would pay to accept the $\color{green}{1/1,000}$ risk of lost film, the implication is that you value the film at £10,000.
I know that $ \dfrac{10}{ \color{green}{ \dfrac{1}{1000} } } = 10,000$, but would someone please explain how and why this is how to determine the value of the film? What's the intuition? I don't perceive the 'implication'; this isn't a math book so am I missing something easy and trivial here? Another example from p 316, supra,
...willingness to pay” studies... begin by trying to determine how much people value their lives by looking at how much they are willing to spend to reduce small risks of death. Suppose, for example, that an airbag for an automobile costs £300, and suppose it is known that every 10,000 purchases of an airbag saves a life. In effect that means £3 million will be spent (by 10,000 consumers) to save that life. Put differently, each purchaser evidently is willing to spend £300 to obtain the benefit of the 1/10,000 chance that it will save his life—and this suggests that each values his own life at £3 million.
The intuition is that your payment $x$ should be the same as the expected loss $E(L)$. With $p$ = probability of loss $l_0$, and a $(1-p)$ probability of losing nothing, this gives
$$ x = E(L) = pl_0 + (1-p)0 = pl_0$$
Solving for $l_0$ leads to your formula $$l_0=\frac{x}{p}$$
Of course insurance doesn't really work like that because there are lots of administrative costs, profits to be made, etc, so the premium is typically more than the expectation of loss.
More importantly, people don't value insurance on expected loss; rather they value it on utility. When I was a student, I paid to insure my possessions. But now I've been in employment for some years, I don't insure my possessions, even though they're worth more and they're less at risk (so I could get a better deal), because I judge that I could afford to replace them. I still insure the building I live in, because regardless of the value of the insurance, I couldn't bear the loss.