Since Mr. Bengen’s original paper was published, the 4 percent concept has been replicated, expanded, criticized and even refined by Mr. Bengen himself. (By using a more diversified portfolio, he later raised the rate to 4.5 percent.)

Critics of the rule point out that it is based on conditions in the United States during a very specific time in history; it also doesn’t take into account items like investments costs, taxes, different time horizons or the reality that most retirees don’t spend their money in a linear fashion. Some people may want to spend more early in retirement and may be willing, even comfortable, making cuts when the market plunges once again. And if retirees want to leave money to their children, they may need to trim their spending further.

Sorting all of this out, particularly without a cushy pension to fall back on, is a complicated task, even for a numbers-savvy retiree. Still, the original 4 percent rule persists as a starting point, and some retirement experts are still comfortable suggesting similar withdrawal rates, with some caveats and new twists of their own.

In a recent analysis, Mr. Pfau compared several withdrawal strategies in an attempt to illustrate how spending patterns might change to guarantee that a portfolio will last for 30 years, even if low rates persist or retires face some other awful combination of events.

He found that people who spend a constant amount adjusted for inflation — similar to the 4 percent rule — would have to reduce that rate to 2.85 to 3 percent if they wanted assurance that their spending would never have to dip below 1.5 percent of their initial portfolio (in inflation-adjusted terms).

So a retiree with $1 million could securely spend nearly $30,000 annually for 30 years, in the best and worst of market conditions. The big drawback, though, is that if economic conditions are generally average, retirees would be left with $794,000 in unspent money. If they were unlucky and experienced terrible market conditions, they would be left with $17,900.