It is in Chapter Sixteen that you need to turn to find the answer to this question. This of course is the long-awaited confrontation between John and Mustapha Mond. When John asks why he will not let his people have Shakespeare and other prohibited classics, Mond explains his reasoning:
"Because our world is not the same as Othello's world. You can't make flivvers without steel--and you can't make tragedies without social instability. The world's stable now. People are happy; they get what they want, and they never want what they can't get. They're well off; they're safe; they're never ill; they're not afraid of death; they're blissfully ignorant of passion and old age; they're plagued with no mothers or fathers; they've got no wives, or children, or lovers to feel strongly about; they're so conditioned that they can't help behaving as they ought to behave. And if anything should go wrong, there's soma."
Thus, according to Mond, classics are prohibited because this brave new world is fundamentally different from previous society. The world has changed to such an extent and humanity has changed to such an extent that the jealousies, passions and loves in works like the plays of Shakespeare would be as anachronistic to the population of this dystopian society as Neanderthal grunting would be to us. They simply would not be able to understand the first thing about works of literature that hark back to a completely different form of society.