Why should the statistical value of life exist? In areas such as insurance pricing and government policy analysis, it is often necessary to assign human life a monetary amount in order to compare it with other monetary amounts.  So economists have a measure called the statistical value of life, which in some sense quantifies how much a person values his own life.  It's usually calculated to be about 10 million dollars for most people.  Now this isn't literally the dollar amount a person puts on his life, because that amount is usually infinity; it's possible that no amount of money would convince the average person to give up his own life, and the average person would be willing to spend any amount of money to save his own life.  So the technical definition is trickier: the statistical value of a person's life is the dollar amount $X$ such that for all probabilities $p$, or at least all values of $p$ relatively close to 0 the person would be indifferent between a situation where their chance of dying is $p$, and a situation where their chance of losing $X$ dollars is $p$.  (An equivalent definition can be given in terms of reducing your chance of death and gaining money.)

My question is not about why this concept is useful; I understand its utility.  (No pun intended.)  My question is, why should the statistical value of life exist at all?  That is to say, why should there exist a single value of $X$ that satisfies this definition for all values of $p$, or even all values of $p$ that are sufficiently close to $0$?

Let's discuss this more formally.  Let $A$ is the set of possible preferences, and let $G(A)$ be the set of "gambles" or "lotteries" over $A$.  Then the von Neumann-Morgenstern theorem states that if a person's preference ordering over $G(A)$ satisfies certain rationality axioms, then the person's preferences can be represented by a utility function $u: A → ℝ$. That means that the value that a person puts on any lottery $L$ is the expected value of$u$ under the probability distribution of $L$.

So I would not be surprised at all if a person was indifferent between a 1 percent chance of getting 10 dollars and a 1 percent chance of getting a chocolate sundae, and was also indifferent between a 2 percent chance of getting 10 dollars and a 2 percent chance of getting a chocolate sundae; that would just indicate to me that the person's preferences satisfy the von Neumann-Morgenstern rationality axioms.  But I don't understand why, if a person was indifferent between a 1 percent chance of losing 10 million dollars and a 1 percent chance of dying, they would necessarily also be indifferent between a 2% chance of losing 10 million dollars and a 2% chance of dying.  That's because living and dying do not comport with von Neumann Morgenstern axioms; the average places the utility of survival at infinity, and yet they assign finite values to small risks of dying.  So I see no reason why lotteries involving risks of living and dying should obey the von Neumann-Morgenstern axioms.

And yet empirically, it seems that studies have found that the statistical value of life is a well-defined and measurable quantity, at least for sufficiently small values of $p$.  So what is the reason for this?  What is the reason that lotteries involving small risks of dying obey the von Neumann-Morgenstern axioms, when living and dying themselves do not?
You asked: 


  why should there exist a single value of $X$ that satisfies this definition for all values of $p$, or even all values of $p$ that are sufficiently close to $0$


There isn't such a value. I would hope that no one claims that there is.

The statistical value of life is a (somewhat lazy) calculation of convenience. Lots of business case protocols need a value for anything that's going into the business case. Changing the probabilities of survival is an outcome of many interventions for which the decision-makers have insisted on business cases, so some method is needed to value these probabilities.

One of the earliest ways to do this, back when relevant research was scarcer than it is today, and computational power was much more limited, was to assign a single value of life, which was calculated using methods that assumed a priori that there existed a single value of $X$ that was an adequate approximation for all values of $p$ that are sufficiently close to $0$.

That method is still used today largely due to institutional inertia.