# Private ownership with overlapping exclusion rights

The previous section showed that a single upstream profit-maximising owner with the right to exclude produced an efficient outcome. Now consider a slightly different legal rule: there are now multiple upstream owners, each of whom can charge downstream users a price *Pj* for the right to extract the resource. Each owner can exclude those who do not

pay this fee. Let us analyse the incentive and efficiency effects of this legal rule. Each user once again compares his marginal private benefit with his private marginal cost. Each user will again find extraction worthwhile as long as:

where now *P* = ^” *P _{i}.* The inverse aggregate demand curve is now:

We can also write this demand curve as *N = N(P).* Now each upstream owner's revenue (and profit) from selling usage rights depends on the price charged by other owners:

Let us compute the Nash equilibrium configuration of prices. Choosing the price to maximise this profit, taking the other owners' prices as given, yields:

Since this holds for each owner i, we can add up over all *n* upstream owners to get:

But

Therefore, letting *N°°* be the number of users under this legal regime, we have:

So:

*Figure 7.3.5* The tragedy of the anticommons

Where the inequality follows from the fact that both sides cannot be equal to zero (otherwise the first order condition for profit maximisation would be violated). But since at *N°°* we have *B'(N°°) >* 0, this means that *N°° < *N*. There is *underexploitation* of the resource. The welfare loss is shown as the shaded area in Figure 7.3.5.

Why is private upstream ownership with multiple excluders inefficient? The reason is straightforward and again involves externalities: if owner *i* reduces his price (whilst other owners hold their price fixed), this reduces the aggregate price faced by downstream firms. This increases the number of downstream users (that is, the aggregate quantity demanded). Thus, as a result of owner i's price reduction, all other owner/excluders enjoy a revenue increase - but without having to increase their own prices. In other words, each price reduction by *i *imposes an external benefit on the other owner/excluders.

Conversely, each price *increase* by *i* reduces the revenue of other owners and imposes a negative externality on them. Since each owner's price reduction imposes an uncompensated positive externality, there will be too few price reductions from an efficiency point of view. Alternatively, since each owner's price increase imposes an uncompensated negative externality on other owners, there will be too many price increases. For both reasons, prices will be too high from an efficiency point of view.

Note also that the inefficiency gets worse as number of excluders rises. Differentiating the first order condition for profit maximisation with respect to *n* yields.

204 *Law and Markets *So:

The aggregate price increases as the number of upstream excluders increases. In the limit, as the number of excluders goes to infinity, we have:

and *B(N ^{00})* approaches 0. As the number of excluders becomes very large, the number of downstream users N approaches zero and the total price P approaches

*B*'(0), gradually choking off all downstream demand. Aggregate benefits also approach zero. In other words, in the limit, the welfare loss in this equilibrium approaches the

*same*welfare loss that occurs under the open access legal regime examined earlier.

One solution to this underexploitation problem is for the multiple excluders to try to collude to keep each of their prices at a level that encourages the efficient number of entrants. This would require setting prices so that:

so that, once again, the aggregate entry price is equal to the external cost at the optimum. However, even if such an agreement was reached, it would not be stable against cheating: if each excluder believes that the others will stick to the terms of the agreement, each will still have an incentive to increase their price in order to increase their own revenue at the expense of others losing revenue. Thus, each excluder still has an incentive to impose a negative external cost on the others by increasing their individual price.

The above analysis illustrates the symmetric nature of the welfare losses associated with multiple usage rights and multiple exclusion rights. Although economists have long analysed and been concerned with the tragedy of the commons, the welfare losses associated with overlapping exclusion rights and the anticommons can be just as serious - perhaps more so, depending on the situation.