WILDLY implausible faux news stories appear each April Fool’s Day, some of which are taken seriously. This year’s clear winner was the National Public Radio feature about a preschool’s new requirement that all applicants submit DNA profiles. As the segment begins, the host Guy Raz is greeted by Rebecca Unsinn, described as headmaster at a school called the Porsafillo Preschool Academy, located in a striking I. M. Pei-designed building in a leafy enclave on the Upper West Side of Manhattan.  Dr. Unsinn walks Mr. Raz through gleaming computer labs where toddlers master C++. She proudly describes the school’s Mandarin Chinese immersion program. We are also told that Dr. Unsinn, a pediatric neurologist, was recruited to oversee the school’s new genetic tests, designed to help winnow 12,000 applications for 32 available spots in next year’s class. As she explains, “We now know that simple DNA testing can determine whether a child will end up at Yale or at Yonkers Community College.” It’s a preposterous claim, of course, but some took it seriously. One Web site recounted the tale this way: “Parents being what they are today, the applicants to Porsafillo don’t even blink at the requirement, some going so far as to have the sample for the DNA test taken while the child is still in the womb, according to an NPR story.” Such reactions bespeak how bitterly competitive the battle for elite preschool slots has become in New York and elsewhere. According to the invisible-hand theory of Adam Smith, greater competition generally promotes better outcomes for society.  But not always. In particular, when buyers are competing for high-ranked positions in some hierarchy, the invisible hand often fails spectacularly. In education, the problem is that a quality is inherently relative. A good school is one that compares favorably with other schools in the area. Although a school’s quality depends on much more than the size of its budget, additional resources can obviously help.  And given the vast sums of money that some parents now have at their disposal, an explosive — and largely unproductive — arms race is inevitable. Tuition at elite preschools has been soaring,  sometimes to much more than $30,000 a year and occasionally within sight of $40,000 — or above that of some prestigious colleges. Yet excess demand persists, and jockeying for acceptance is the subject of much New York lore. In one notorious episode a decade ago, Jack B. Grubman, then a telecommunications analyst for Citigroup, told a friend in an e-mail that his boss, Sanford I. Weill, then Citigroup’s chairman, helped to secure spots in an exclusive Manhattan nursery school for Mr. Grubman’s twins after he began recommending that investors buy AT&T stock. (Mr. Weill said he put in a good word for Mr. Grubman only because he was a valuable employee.) Mr. Grubman’s children attended the school, and Citigroup pledged $1 million to the organization that runs it, the 92nd Street Y. In the e-mail, Mr. Grubman described the school as “harder than Harvard” to enter. WITH such stories in memory, it’s hardly surprising that many failed to spot the NPR report as a joke. The elite preschool market illustrates a simple but important truth: Contrary to the slogans of champions of minimal government, we don’t always get the best possible outcomes when people are free to decide for themselves how to spend their hard-earned money. When all wealthy parents raise their bids for slots in elite preschools, they succeed only in bidding up tuition levels. There are still no more slots than before. Most societies take at least some steps to curb waste that results from arms races in education and other domains.  Consider the practice of kindergarten redshirting — so called because of its resemblance to the practice whereby universities hold athletes out of competition during their freshman year so they’ll be bigger and stronger during their four remaining years of eligibility. Ambitious parents might consider redshirting their kindergartners, because they would then be older, smarter and more emotionally mature than their classmates.  And because school performance is graded on a curve, they would be more likely to win admission to an elite university.  But the same option, of course, is available to other parents, and if all took it, no child would perform better in relative terms.  We’d just end up with an older crop of kindergarteners. That’s why parents have good reason to favor the laws in most jurisdictions that take the kindergarten start date out of their hands. It would obviously be a much more radical step to impose limits on how much parents could spend on private schools.  But what if we adopted the Buffett Rule, under which top earners’ tax rates would be no lower than those paid by middle-income families? That would reduce what top earners could bid for the scarce things they want.  But because the allocation of elite preschool admissions, penthouses overlooking Central Park and other such prizes is settled by relative bidding, the question of who gets what would be unaffected. So with no real sacrifice, the rule would generate new revenue for reducing deficits and rebuilding tattered infrastructure. Top earners, meanwhile, will continue to reap the bulk of all income growth, and the preschool admission battle will grow steadily more intense. The only reason that no elite preschool has adopted DNA tests is that no one has figured out how to use them to predict academic achievement.  Yet. Robert H. Frank is an economics professor at the Johnson Graduate School of Management at Cornell University.