In this Saturday, July 4, 2009. file photo, U.S. Vice President Joe Biden, right, talks with his son, U.S. Army Capt. Beau Biden, at Camp Victory on the outskirts of Baghdad, Iraq. On Saturday, May 30, 2015, Vice President Biden announced the death of son, Beau, from brain cancer. (Khalid Mohammed/AP)

“The Stress Cost of Children.”

It’s all in the headline — or, in the case of academic papers, the title — and this paper, released recently by the National Bureau of Economic Research, somehow was more alluring than its competitors coming to my attention: “Does Exporting Improve Matching? Evidence from French Employer-Employee Data.” Or, “Different Types of Central Bank Insolvency and the Central Role of Seignorage.”

In this season of high school graduations, including one in my own household, the “stress cost” paper turned out to be particularly apt. It examined the impact not only of adding children to a family but also of subtracting them — the Empty Nest, quantified with longitudinal data and analyzed with reference to Lagrangean multipliers and the Ashenfelter dip.

But you don’t have to be adept in multivariable calculus to understand — indeed, to predict — the results. Hielke Buddelmeyer and Mark Wooden of the University of Melbourne and Daniel S. Hamermesh of Royal Holloway, University of London, examined surveys filled out over a decade by couples in Australia and Germany, asking respondents how often they felt pressed for time and how much they worried about finances.

The authors found what any parent — certainly any mom — could have told you.

“[W]e show that births increase time stress, especially among mothers, and that the effects last at least several years,” they write. “Births generally also raise financial stress slightly.” But overall, it was the lack of time, not the lack of resources, that was so draining.

“There is no reasonable transfer of earnings from husband to wife that can compensate for the increased time stress that she experiences with the new child,” the authors report. In one of their simulations, the required one-time transfer to offset the stress would amount to twice the husband’s annual salary.

Economically speaking, the decision to have children is not utility-maximizing. And yet, most of us — intentionally, passionately, joyfully — make this least rational of choices. More than once.

But here’s the more intriguing part of the study: The effect of emptying the nest is much less than that of filling it. “While the departure of a child from the home reduces parents’ time stress, its negative impacts on the tightness of the time constraints are much smaller than the positive impacts of a birth.”

Translation: Children may leave your home, but they never leave your heart. To have children is to permanently devote a segment of your brain to tracking their whereabouts and worrying about their well-being.

I suspect this is why the death of Beau Biden at age 46 felt so gut-wrenching, even to strangers. As parents of young children, we are responsible for their well-being and their safety. We can never protect them entirely, as the car accident that took the life of Joe Biden’s first wife and infant daughter cruelly demonstrated. Yet at the outset we have, or imagine we have, some degree of control: the sleeping on the side (or is it the stomach?); the well-fortified car seat; the preservative-free mashed peas.

The arc of parenting is the process of increasingly accepting the futility of managing risk. You have to let them out into the world — into a car driven by someone else, onto a playground where another child might be cruel, into a classroom where they might stumble. Beau Biden survived the accident; he made it safely through deployment in Iraq; he seemed to have beaten the brain cancer. And then it got him.

Eternal vigilance is the price of parenthood, but it is insufficient. As I write this column, my family is asleep upstairs, awaiting our younger daughter’s high school graduation later in the day. I am exquisitely mindful of how increasingly elusive those together moments will be. Our older daughter is working in New York this summer and studying in London next semester. The younger will be off to college in August, a 2-hour, 40-minute plane ride away.

So I look at the shoes strewn across my front hallway and the pile of dirty dishes in her room and try to leaven my exasperation with wistfulness. Soon enough, I will mourn the absence of mess. (Okay, not really. But I will miss the life force that accompanies the mess.)

Buddelmeyer, Wooden and Hamermesh advise that my “time constraints” will diminish, although not quite to an extent that is statistically significant. “Births tighten the constraints much more than departures loosen them,” they write.

Indeed. We deride helicopter parenting; we vow to avoid it. Yet our hearts cannot help but hover.

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