How much should it cost to close the beach? For most of the past week a private surfing contest has closed off Steamer Lane
- one of California's most popular waves - while paying only a few hundred dollars in permit fees. I think this is far too cheap.

Consider the following thought experiment: If somebody were to set up
at the beach and offer \$1 to each surfer who stays out of water they'd
all decline. However, if the amount were upped to \$100, several would
accept. To me this indicates that the price to close off the beach should be tens of thousands of dollars rather than a few hundred.

Is this a reasonable way to price externalities? Is there some other model where it would make sense to charge only a few hundred dollars for beach closures?
The best way to find out the competitive value of the temporarily privatised beach would seem to be to auction licences, charge a tax based on the self-assessed value for holding them, and require the holder to relinquish them at the self-assessed value. This is basically the idea of Eric Posner &amp; Glen Weyl’s Radical Markets (2018).

Given that the organiser of the contest has to commit to a suitable date (presumably during the right season, at the weekend, etc.) she will be willing to pay a tax for holding the licence based on a self-assessed value high enough to deter other (groups of) surfers from buying out her licence for that particular date. The other surfers will likely have less propensity to pay the assessed value of the contest organiser than to pay nothing for using the beach the following day. 

That said, the public authority would likely need a mechanism for guaranteeing that the beach remains public in principle. This could be achieved by limiting the number of days per year that the beach could be privatised, or requiring qualitative criteria (such as an event) for issuing a licence. In any case the public condition is mutually exclusive with privatising the beach for dwelling.