Hotelling doesn't say what you think it says

Hotelling can also generate complete differentiation

“Why do gas stations locate across the street from each other?” Hotelling.

“Why do all SUVs look the same?” Hotelling.

“Why are all social media video reels the same?” Hotelling.

“Why does every restaurant have a chicken sandwich?” Hotelling.

The Hotelling model is one of the core models of competition in economics. In the world of imperfect competition, it’s Bertrand, Cournot, and Hotelling as the big three. That is the usual takeaway from Hotelling: sellers converge. It is certainly what I emphasized when teaching Econ 101. In the original Hotelling model, differentiation is often described as “location.” Where should I locate my shop when people have to come to buy stuff? But you could generalize that to be something like product type or quality in a more abstract space.

That’s fair for Econ 101. We can go a bit deeper at Economic Forces. I’ve been thinking a lot about Hotelling while updating my paper on the waterbed effect, so let’s talk.

In general, there are really two forces going on in the Hotelling model. You can pick a location, but you can also pick a price, and the interaction between those two pushes against each other. Moving further away allows you to increase price through differentiation. Moving closer allows you to capture more consumers, but you have to lower your price to win those consumers. Let’s call that the differentiation force, which is a reason for the sellers to differentiate from each other. That force allows them to raise the price and is a reason for the sellers to move away from the middle.

Let’s work through when each force dominates.

The classroom model

As is usual, the textbook version strips a few things down. Take, for example, McCloskey’s Applied Theory of Price. In that book, she teaches, like other textbooks do, that you have two sellers picking spots on a road. That’s it.

Imagine a beach with consumers spread out from one end to the other. There are two ice cream carts that charge the same price for their ice cream. They sell the same ice cream, and they have the same sort of cost for their ice cream. Consumers simply walk to whichever cart is closer in this situation.

Suppose one cart stands a quarter of the way down the beach, and the other stands 3/4 of the way down the beach. The consumer at the midpoint is indifferent, and each cart serves half the beach.

Now move the left cart towards the center a little bit. It keeps all of the consumers it was already winning on the left, but it wins some of the consumers on the right, so it wants to. Same thing with the ice cream cart on the right: if it moves a little toward the center, it wins more customers and doesn’t lose any. That logic keeps pulling the carts together until they both reach the midpoint. This is the convergence of Hotelling’s Law, or the “principle of minimal differentiation.”

This is also why you get the median voter result. There are no explicit prices, and so you move to the middle, to the median voter, under similar assumptions of one dimension etc. You break that result in a variety of ways, but one way is for the candidates themselves to have a preference, and that then becomes like a cost, pulling them the other direction, the other force pushing them away from the median. We need to have some opposing force, which is absense in the first past Econ 101 model.

Instead of pure location, you may have seen alternative “Hotelling model” which fixes the location and asks how firms are going to price. This is the type I’ve been working on recently, starting with the waterbed paper I wrote about in this newsletter.

In that model, as far as you’re concerned as a seller, you just face some demand curve that you back out from the locations and pricing of the other seller. There, by assumption, there is no convergence, but firms do think about the trade-off of pricing.

Hotelling had both

But neither of those is “Hotelling” in the original sense. Hotelling’s 1929 paper didn’t have this fixed-price location game. He allowed the sellers to choose prices and then used the resulting profits to study where they would locate. He had a fuller model of pricing and location, this two-dimensional competition.

Unlike the classroom example, Hotelling wasn’t tipping the scale towards convergence. He had this interaction between these two forces pulling in opposite directions, where moving further away from the center would allow you to increase prices through differentiation.

Hotelling worked backwards to solve his model. He took the sellers’ locations as given and calculated the prices that they would charge. He then fixed the location of one seller, say A, asked the other seller, B, where it would locate, and vice versa. Kind of a crude way of trying to construct an equilibrium but it was 1929.

As B moved closer to A, Hotelling’s calculation said that B would attract more customers and charge a higher price. Moving closer intensified competition with A but also placed B between A and the larger body of consumers. That stiffer competition created by proximity was always outweighed by the larger consumer base.

Hotelling assumed that travel costs were linear, with additional miles equally painful, regardless of where the consumer starts. That creates a constant force, whether you move from the far left a little bit or from roughly the center all the way to the center. That travel cost that the consumer is giving up is the same. linear costs limit the benefit from differentiation. In Hotelling’s model, the convergent force with that constant fixed level of linear travel cost always dominates.

So we get convergence.

Fixing Hotelling

First, it’s important to note there was an issue in Hotelling, as with many mathematical papers from that time period. To make the problem tractable, Hotelling assumed that a small change in price would produce a small change in demand. That may not be true in general. You can have discontinuities, like in a Bertrand model. It took 50 years but this was ultimately made explicit by Claude d’Aspremont, Jean Jaskold Gabszewicz, and Jacques-François Thisse, although Hotelling noticed it in a footnote but dismissed it.

They also clarified the forces further. Instead of linear cost, they replace that with quadratic costs before allowing the firms to choose prices. This means the marginal tradeoff from moving from 1 mile to 1.1 miles is different from 2 to 2.1. Now, with this growing mismatched location travel cost, the balance between reaching more customers and insulating yourself from competition can change depending on where you’re going. They show that if you have quadratic costs, the equilibrium is for the sellers to move as far away as possible to the edge of the model.

They have a Hotelling location model that produces the exact opposite outcome of what we teach in the classroom. You get maximum differentiation. It’s Hotelling. People are choosing location. It’s also Hotelling in that they’re choosing prices.

You can see this logic by thinking of moving the left firm a little inward from the endpoint. In the fixed price model, that seller wins customers, as we said, and they don’t lose anyone else. In the price-setting model with quadratic cost, it will still win customers, but in order to do that, it’s going to lower its margins to win those customers. At the endpoint, under their normalization, the loss from the lower margin is twice as large as the gain from new customers, and the inward move is going to lose any money. The exact magnitude is arbitrary, but it shows that at the edge, you don’t have an incentive to move in.

Is it really Hotelling?

Have I ever said that there are tradeoffs? Well, there are tradeoffs here. In general, you are going to want to move closer to your competitors sometimes and move away at other times, and the equilibrium will sort out what happens overall.

In updating that waterbed paper to expand to the big 3 models (spoiler: waterbed doesn’t cause harm in any of them), I realize these models are all just instantiations of the same trade off. The Economic Forces are really close to the traditional Bertrand forces: lowering your price wins consumers but reduces the margin of every unit you sell. Similarly, in a Cournot model, producing more wins sales but pushes down the market price. The fundamental competitive force in all of these is this trade-off between quantity and margin when maximizing profit.

I think we need to be careful about what we mean by location competition/Hotelling competition. It’s not enough to see two things that are similar and conclude that it’s Hotelling, because the exact same model under different specifications shows that that would predict complete differentiation. There’s an analog to how we think about similar prices. If we see two firms with the same price, some people will want to say that that’s collusion, but that’s also what you’d predict out of competition. It’s underspecified.

Go back to the chicken-sandwich example, which came from Byrne Hobart. Is that Hotelling, or is that a general shift towards chicken consumption in the model overall? What if there is more chicken now? That could be a combination of supply and demand reasons. Maybe chickens became cheaper to raise. Chicken availability (neither supply nor demand, but it’s what is measured) per person has been rising since the 1940s, passed pork in 1996, and passed beef in 2010. That seems relevant to things and not clearly about Hotelling competition.

To tease out the more Hotelling-like story, I’d want more detail on how one restaurant’s decision affects how the other will respond. Once Popeyes has a chicken sandwich, McDonald’s will lose customers who want chicken unless it adds one too. Sure, the relevant evidence wouldn’t necessarily be that McDonald’s added a chicken sandwich, but we should expect places that are facing more competition from Popeyes to respond more quickly. For example, did places closer to Popeyes, or business franchises with more stores near Popeyes, respond more quickly? That might be some evidence. But it’s speculative.

The important thing is that there’s always a trade-off between trying to attract more customers (in this case, by moving closer to them) or, in other cases, by lowering the price to attract them, versus making money on the customers you kind of already have. That shows up across all big three models of imperfect competition.

The key insight of Hotelling is that the force to conform, to be more like the median consumer, is there. That force alone is going to pull products toward each other, and we should take that force seriously. The takeaway is not that this is the only force, so they’re actually in practice pulled all the way to minimal differentiation. Multiple margins of adjustment.