I will try and build a small model in which it is true that “charging a perhaps even substantial price for beach access would be welfare improving for **all** potential beach goers”, a claim I made in my last post. In this post I will take a first few steps in this direction, first only demonstrating my claim that beaches potentially suffer from the “tragedy of the commons” before I will tackle the main question in the next post.

By the way, the interested reader may want to look at the literature on the economics of clubs for more on this topic. A good starting point may be “Clubs” by Suzanne Scotchmer, 2008, in the New Palgrave Dictionary of Economics also available here.

So what do we need in this model? We need potential beach goers and we need to think about the benefit that these beach goers derive from going to the beach. We already have a lot of options here. We could have a finite number of potential beach goers or we could think of them as a continuum of beach goers. The first assumption is obviously empirically correct but the latter may be more practical when we are thinking of a lot of beach goers. Let me here start with having a finite number of beach goers, but I might change this later. A beach goer is now assumed to derive a “utility” from going to the beach (versus pursuing his or her second best alternative) that is a function of how many other people there are on the beach. Let us call this function , where is the number of other people on the beach (excluding the person whose utility we are here looking at). Of course, in reality different potential beach goers have different such utility functions, and of course, people do not really have such a clear function in their mind at all. But people are probably more or less happy being on the beach with more or fewer other people on the beach and people probably at least to some degree make their choices to which beach they go dependent on their expectation of the number of other people on the beach. More worrying than our assuming the existence of such a utility function is our assumption that all people have the same utility function. This is almost surely wrong, although it is actually not so easy to assess this empirically. I will assume that all people have the same utility function for the moment, but we should keep in mind that this is most likely wrong. We may want to come back to this question at the end.

Now to the shape of this utility function. I would assume that it is ultimately decreasing in and eventually negative for sufficiently large . As I am not so interested in beaches that do not suffer from an overuse problem I will simply assume that the utility function is decreasing for all and of course positive for .

With the model as it is so far I can now replicate (or demonstrate) my argument of the previous post that beaches can be inefficiently overcrowded. Suppose that the number of potential beach goers, call it , is such that is negative. Given this, how many people will go to the beach?

In an equilibrium (call it Nash equilibrium if you like – as what I described here is really an n-player game), we expect essentially so many people, call it , to go to this beach such that . Why? Well if fewer people than go to the beach then a potential beach goer who is not on the beach would derive a positive net utility (over the second best alternative) from going to the beach. So she should go. And more people will come until we have people on the beach. If more people than are on the beach, people on the beach will suffer a negative utility and will start leaving until the remaining number is again than .

Of course in reality we do not exactly expect people to go to the beach for various reasons. One possible reason is that our model is simply not a 100% accurate description of the real world. But even if our assumptions about people’s utility functions were completely correct we still have made an implicit and quite radical assumption about what people know about the number of people on the beach when they make the decision whether or not to go to the beach. In reality when you make this decision you do not know how many people are on the beach already and how many people will still come later. Also, once you have driven perhaps a fairly long way to the beach and then you see that it is rather crowded you may decide to stay even if, had you known about this before you left, you would not have driven to the beach in the first place. However, as I argued in my previous post, the approximate number of beach goers at various beaches is often roughly commonly known. People who live in the area have a pretty good idea about these numbers and tourists can also inform themselves fairly well from their respective hosts. This information is to some extent typically also available on the internet. On any given day you might find that you were lucky and that the actual number is somewhat lower than expected or unlucky and the actual number is somewhat higher, but on average the numbers are not so far off from what people expected.

Now back to the equilibrium number of beach goers on our beach, , what have we learnt from this simple analysis so far? Well, we have the tragedy of the commons in a nutshell. Despite the fact that all potential beach goers would derive a potentially high extra benefit (over their second favorite activity) from going to the beach – if only the beach is not overcrowded – the equilibrium number of beach goers is such that everyone is just indifferent between going to the beach and doing their second favorite thing (staying at home, for instance, or going to another beach).

If somehow we could cap the number of beach goers at some lower level, say , for which by assumption , we could improve the utility of all the people who are allowed to go to the beach without hurting those who are not allowed. This is because without the cap the latter group would have been indifferent between going to the beach and their second favorite thing anyway. One should probably now re-examine the assumption that all people have the same utility function. I will leave it to the reader at this point, but will tackle this in my next post when, I hope, I will finally demonstrate how it is possible to impose a cap through a price for beach access and how this can be welfare-optimal.

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There is a problem with the argument and this problem nicely motivates why one might want to work with a model with a continuum of agents. It is not quite true that in this model that in every (pure strategy) equilibrium, everyone would end up indifferent between going to the beach and not. Suppose first that there is a single beach (the argument generalizes) and that there is an equilibrium in which n>1 people end up going to the beach and ending up with the outside utility 0 due to the crowding on the beach. In that case, n-1 people going to the beach would be an equilibrium too. Everyone ending up going to the beach will be better off than by not going to the beach, and everyone who would switch to going to the beach would be no better off. If there is no positive integer n such that u(n)=0, then every pure strategy equilibrium would have the property that everyone at the beach would be better off than by not going to the beach. Indeed, it must be the case in equilibrium that everyone at the beach must get non-negative utility and all achievable non-negative utilities are by assumption positive. This shows that whenever an equilibrium in pure strategies exists, there exists an equilibrium in pure strategies in which some people are better off on the beach than at home. The argument doesn’t really depend on there being a single beach. Lastly, it would be of course vacuously true that in every pure strategy equilibrium everyone receives their outside utility if there would be no pure strategy equilibrium at all. But there always is: Assume there are k beaches and n people. Put the first person on the first beach, the second person on the second beach, and continue until you run out of persons or empty beaches. If the utility of two people per beach is still positive and you haven’t run out of people, put yet another person on the first beach, and continue the obvious way. You will end up with an equilibrium in pure strategies.