Now, what if the firm is not paying the law clerk at all. It would seem odd that the firm could claim the client has to pay certain value for "overhead" when in reality there is no overhead. On the other hand, since the law clerk is not a lawyer, can the firm charge the client for the value of the time of the law clerk?
My guess is these questions have probably been addressed by ethics opinions, but I have not done the research.
What I can tell you is that there is a new opinion (available here) out of the New York State Bar Association that holds that a "law firm may bill a client for work performed by a student-intern despite the fact that the law firm does not pay the intern, because the intern receives academic credit for the work, as long as (i) the internship program complies with applicable law, (ii) the educational institution does not object to the client charges, and (iii) the charge is not excessive."
Not everyone agrees this is the correct decision. See this article in Above the Law for a negative review of the opinion.
UPDATE (8-26-16): The ABA Journal online is reporting that various law student groups and some labor groups have signed a letter to the New York Law Journal calling the ethics decision “fundamentally flawed” and asking the ethics committee to reconsider.
UPDATE (9/6/16): Legal Ethics in Motion has an update here. It states, in part: "In response to the opinion, several organizations wrote an open letter, printed in the New York Law Journal, criticizing the decision as “fundamentally flawed.” . . . In the letter, the signatories ask the ethics committee to reconsider its decision because the opinion “fails to consider the circumstances of most unpaid legal internships and the important moral questions they raise.” The letter also challenges the assumption that unpaid internships at private firms comply with the applicable labor laws. The organizations contend that when a law firm charges for an intern’s free labor, they implicitly derive a substantial and economic benefit that cannot be offset by the academic credit that the interns receive, and they therefore may be entitled to pay."