Efficiency principles are increasingly called upon to study features of human language and communication. Zipf's law of abbreviation is widely seen as a classic instance of a linguistic pattern brought about by language users’ search for efficient communication. The “law”—a recurrent correlation between the frequency of words and their brevity—is a near-universal principle of communication, having been found in all of the hundreds of human languages where it has been tested, and a few nonhuman communication systems as well. The standard (...) explanation for the law of abbreviation derives from pressures for efficiency: speakers minimize their cumulative effort by using shorter words for frequent occurrences. This explanation, we argue here, fails to explain why long words exist at all. It also fails to explain why the law of abbreviation, despite being robust, is systematically weakened by many short and rare words. We propose an alternative account of the law of abbreviation, based on a simple cultural evolutionary model. Our model does not require any pressure for efficiency. Instead, it derives the law of abbreviation from a general pressure for brevity applying to all words regardless of their frequency. This model makes two accurate predictions that the standard model misses: the correlation between frequency and brevity is consistently weak, and it is characterized by heteroskedasticity, with many short and rare words. We argue on this basis that efficiency considerations are neither necessary nor sufficient to explain the law. (shrink)