Alex Tabarrok writes

The conclusion seems right to me - file-sharing increases social-welfare, so in theory a win-win solution is possible, but in practice the increase comes at the expense of music firms.

That reminded me of Zimran ("winterspeak") Ahmed's back-of-the-envelope estimate that the music

copyright tax causes a deadweight loss of $3 billion per year.

Longtime readers may remember that I mentioned Zimran's analysis here, and I probably did in earlier posts as well, but they are not as well archived (I was doing my own archiving when I first started blogging).

A back of the envelope accounting of my own:

Termination of a copyright benefits the public, but destroys a privilege that has value to the rights holder. It thus seems fair to terminate the copyright if the public has given the rights-holder something equal in value to the copyright, but not before.

We can call the value of the copyright the present value of all future rents that could be extracted from the copyright, and the public's compensation to the rights-holder the present value of all past rents extracted from the copyright.

If one will (with apologies to Joel Cohen) accept the hypothesis that sex is a sublimation of the mathematical urge, and on those grounds excuse the following mathematics as adult entertainment, then if at time t rents r(t) can be extracted from the copyright, then the present value of the copyright at time T is

/inf

exp[iT] | r(t) exp[-it] dt

/T

and the present value of all past rents (the public compensation) is

/T

exp[iT] | r(t) exp[-it] dt

/0

where i is the interest rate used to discount future value. It is fair to terminate copyright once these two expressions are equal, and the neatest way I know to express this mathematically is to define R(t) as the year-zero value of a t-year copyright:

/t

R(t) = | r(t) exp[-it] dt

/0

and then it is fair to terminate copyright after t years if R(t) = ½ R(infinity). To put it non-mathematically, if we grant a copyright worth half of a perpetual copyright, then when the copyright expires, the public will have given the rights-holder rents whose present value is equal to present value of the rents "lost" through not having a perpetual copyright. Or put another way, a copyright of this duration splits the value evenly between the rights-holder and the public.

Translating this idea into a practical policy recommendation requires estimating r(t), which is approximately the sales curve of the work. Now most works sell well initially and then decline into obscurity, but let's be generous and consider a work of enduring value, whose sales curve is constant through time. In this case, the above condition is met after .693/i years. If two percent is the interest rate used to form policy, this recommends a flat 35-year copyright term for works of enduring value. The "proper" term for ephemera is of course shorter.

So ... whenever the producer sets the price of something above its marginal cost, it's a tax?

Robert writes:

Most works do not sell well initially. In fact, most works don't sell at all. Does that effect your estimate?

Most works do not sell well initially. In fact, most works don't sell at all. Does that effect your estimate?

The absolute number of units sold is irrelevant; only the relative shape of the sales curve over time matters. A valuable work is valuable because the public is willing to pay more to use it, but over time, the public has already paid more to use it.