Except that this appears not to be the case. These chaps (Chen & Riorden) suggest that specific conditions (duopoly and oligopoly) exist where the addition of new competition can result in prices increasing rather than reducing.
In a discrete choice model of product differentiation, the symmetric duopoly price may be lower than, equal to, or higher than the single-product monopoly price. Whereas the market share effect encourages a duopolist to charge less than the monopoly price because a duopolist serves fewer consumers, the price sensitivity effect motivates a higher price when more consumer choice steepens the firm's demand curve.
If the addition of more law college places does not create an effective competitive market (i.e. the existing oligopoly is sustained albeit with more firms) then it is as likely that existing suppliers will increase their prices in response to new competitors. The question then becomes a matter of what it is that the consumer is buying.
For law schools, the answer has always been that the elite private schools offer a better chance of preferment post-qualification. In a more competitive market – one where more lawyers are trained but there are no more placements – then the premium for the elite schools is clear and justified. What students buy isn’t a better education but an increased chance of getting the placement that provides access to future success.
The problem here isn't the supply of legal training but the supply of places where that legal training is applicable. And the "who you know" principle gives an advantage to established, successful and elite institutions. And they will cash in on that advantage.