# Economics on the beach IV: welfare optimal pricing – a model

This post builds on the previous two, economics on the beach II and economics on the beach III. I have started this, so I need to finish this now. In this post I will finally try to 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”.

Here is a first attempt. Let me assume that all potential beach goers have the same utility of going to the beach (over their second best activity) as a function of the number of other people on the beach, $\displaystyle u(k)$. We now need to introduce prices. It seems a safe assumption that, ceteris paribus as the economists like to say (meaning all else the same), people prefer to pay less over more. In principle we could work with any final utility function $\displaystyle v(k,p)$ that depends on both the number of other people $\displaystyle k$ and the price $\displaystyle p$ as long as it is decreasing in both arguments. We do not lose much, and it is (a little bit) easier to understand, if we use $\displaystyle v(k,p) = u(k) - p$.

Now fix any positive price $\displaystyle p$. What would the new equilibrium number of beach goers be? By the same argument as in the previous post, and with the same caveats, we now expect a number of beach goers, call it $\displaystyle \hat{k}(p)$, that makes everyone indifferent between going to the beach and doing their second favorite thing. In other words, $\displaystyle \hat{k}(p)$ must be such that $\displaystyle u(\hat{k}(p))=p$. And as $\displaystyle u$ is decreasing in $\displaystyle k$ we have that the higher the price $\displaystyle p$ the lower is $\displaystyle \hat{k}(p)$ – the fewer people are on the beach.

Are people more happy now having to pay for beach access? No. However, there are also not less happy. Why? This is so because the higher price does two things. First, the people who go to the beach now have to pay this price, which they do not like. But second, there are now fewer people on the beach, a fact that they do like. However, on balance, the two effects wash out. So we are back to square one.

But I have not played all my cards yet! In fact I have at least two routes to go. Let me take the less obvious one first, and I will come back to the more obvious one later (what is it?). I have so far assumed that all potential beach goers have the same utility function, an assumption that we agreed, I assume, is not very plausible. Let me now introduce heterogeneity among our potential beach goers. There are at least two ways of doing this. I will assume that people still do not differ in their $\displaystyle u$ function, but in their willingness to pay in order to get some $u$ through accessing the beach. And now, while I could stay with the model with a finite number $\displaystyle n$ of potential beach goers, it strikes me as more elegant and easier to turn to a model with a continuum of beach goers. I think you will see why. Let me take an arbitrary potential beach goer. Her or his utility shall now be given by the function $\displaystyle v(\beta,p) = u(\beta) - \alpha p$, where $\displaystyle \beta$ is now the proportion of all potential beach goers that actually end up going to the beach, $\displaystyle p$ is still the price, and $\displaystyle \alpha$ is a parameter that describes this person’s willingness to spend money. A person with a low $\displaystyle \alpha$ does not suffer that much from paying a high price (I guess this is probably a wealthy person, but could also be just a beach fanatic), while a person with a high $\displaystyle \alpha$ is very reluctant to spend any money in order to get beach access. I can easily introduce heterogeneity now by assuming that there are different people with different $\displaystyle \alpha$. In fact, and that is why a model with a continuum of potential beach goers is now easier, it is easier to assume that a person’s $\displaystyle \alpha$ is distributed according to some continuous distribution. One could of course also work with only a finite number of possible values for $\displaystyle \alpha$, but this is more clumsy in the analysis.

In order to make some calculations I will assume a more specific setting. I will assume that $\displaystyle u(\beta) = 1-\beta$. It is strictly decreasing in $\displaystyle \beta$ and is zero exactly only at $\displaystyle \beta = 1$ (that is when all potential beach goers actually go to the beach). I will also assume that people’s willingness to pay parameter $\displaystyle \alpha$ follows a uniform distribution on the interval $\displaystyle [0,1]$.

Having made all these assumptions (and making the assumption that we shall have an equilibrium in this game with a continuum of players – see my previous post on why I believe equilibrium makes sense in this context), I can now let math take over to work through the consequences of this model.

Now every person is different and different persons make different choices. For a given price $\displaystyle p$, a person with a given $\displaystyle \alpha$ goes to the beach if and only if $\displaystyle 1 - \beta - \alpha p > 0$, or equivalently, if and only if $\displaystyle \alpha < \frac{1-\beta}{p}$. Note that because of the continuum assumption I can ignore people who are exactly indifferent between going to the beach and their second favorite activity. They have zero mass in such a model.

This now means that for a given price $\displaystyle p$ and a given proportion of actual beach goers $\displaystyle \beta$ the actual beach goers are exactly those people with an $\displaystyle \alpha < \frac{1-\beta}{p}$. How many of these do we have? Or, more accurately, what is their proportion? Well this is given by the probability that $\displaystyle \alpha < \frac{1-\beta}{p}$. And, as we assumed that $\displaystyle \alpha$ is uniformly distributed on $\displaystyle [0,1]$, this probability is given by $\displaystyle \frac{1-\beta}{p}$. But this now means that when the proportion of actual beach goers is $\displaystyle \beta$ it will then be given by $\displaystyle \frac{1-\beta}{p}$. So the two (and this is the equilibrium condition) must be the same: $\displaystyle \beta = \frac{1-\beta}{p}$ and we obtain an equilibrium proportion of actual beach goers of $\displaystyle \beta^*(p) = \frac{1 }{1+p}$. In this model, if people are charged a price $\displaystyle p$ to go to the beach a fraction of $\displaystyle \beta^*(p) = \frac{1 }{1+p}$ actually pay this amount and show up at the beach.

We can now do all kinds of interesting things with this model. First, we can verify that if the price is zero, the model leads to the same conclusion as the previous one. Here, everyone goes to the beach, but everyone is in the end indifferent between going to the beach and the second favorite activity and nobody derives an actual positive benefit from being on the beach. Second, we can finally compute the welfare optimal prices as I have promised. This opens a new can of worms, of course. What is welfare? Let me just use, without discussion, what is typically called utilitarian welfare, but you can use your own measure if you like. Utilitarian welfare is simply the equally weighted sum of all people’s utility. In our case the sum will have to be an integral, as we have a continuum of individuals. Taking our equilibrium condition as given utilitarian welfare is given by  $\displaystyle W(p) = \int_{0}^{\frac{1}{1+p}} \left(1- \frac{1}{1+p}-\alpha p\right) d \alpha$, which one can compute to be  $\displaystyle W(p) = \frac{p}{2} \frac{1}{(1+p)^2}$. The reader can verify that the welfare maximizing price, in this model, is equal to one (a trick: maximize the natural log of welfare).

So what do we have? At a price of one (in whatever currency we are working with here), half of all people go to the beach. These are the people with a high willingness to pay. That is, with an $\displaystyle \alpha < \frac12$. These people derive a strictly positive benefit from being on the beach and are therefore better off in this case than under zero prices. The other half of the people does not go to the beach and pursues their second favorite activity. They derive zero extra benefit. So they are not better but also not worse off than under zero prices. Going from zero to positive prices we therefore have what is called a Pareto-improvement, we make some people better off without making anyone worse off.

Could we make all people better off? Yes (and this by the way is the route two I mentioned earlier). I have so far not said where the money that all these people are paying actually goes. Supposing that the group of potential beach goers is an easily identifiable group (the people who live in the area as well as all registered tourists), then the income generated from the beach goers could be paid out equally to all potential beach goers, those who go and those who don’t. In our model all potential beach goers would therefore receive a money amount of $\displaystyle \frac{p}{2}$ regardless of whether they go to the beach or not. Note that this does not change their incentives to go to the beach, unless their utility function changes when given a small amount of additional wealth. But then now everyone is better off under positive beach prices compared to zero prices. The world would be a better place.

The reader may now want to come back to the assumption that all people have the same $\displaystyle u(\beta)$ function. Sufficient heterogeneity here will change some of the insights somewhat and clear Pareto improvements will typically not be possible. But the utilitarian welfare will typically not be maximized at zero prices in such a model either.

Another thing one could do now is to ask how the price is chosen. Perhaps it is chosen by majority voting among all potential beach goers. What price would they vote for? Would it be the welfare maximizing price?