Urban Mathematics and ice cream

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People tend to hear the word ‘Mathematics’ and run away… So here is a simple article that explains, without any scary formulas, the basic economic principle that leads to the creation of cities and suburbs. It starts with a little something called “The ice cream vendors” that illustrates how a free market leads to a familiar and problematic spatial situation.

Imagine a 1000-meter beach strip. On the sandy surface, there are people laying on their towels, spread evenly anywhere on that strip. Now we’ll add an ice cream vendor. In order to serve the customers in the best possible way, the vendor is located exactly in the middle of the beach. This way the maximum distance his customers need to go is 500 meters.

Now, the rumor had spread out and a second vendor came to the beach. Since this is a theoretical question and not a realistic portrayal of Israel, the ice-cream vendors don’t pummel each other’s faces, but rather turn to the lifeguard to resolve the dispute. As our lifeguard studied geometry and economics, he therefore concludes that the ideal solution for the market is dividing the beach into two sections, two 500-meter strips. With each of the ice cream vendors standing at the center of his designated strip. Thus each of them will serve 50% of the market and the maximum distance each customer will have to walk will be 250 meters. The vendors agree and stand where they were told.

But then, one of the vendors noticed, that if he will move towards the center, he would increase his market share. Assuming that every customer tries to minimize the walking distance. By moving towards the second vendor, he increases his share of the coast.

But the other vendor is no fool either and is not going to take this lying down. He quickly reaches the same conclusion, so to earn back his market share he moves towards the first vendor. Not long after, the two of them are standing next to each other in the middle of the beach. None of them is willing to move because it will lead to a loss

So… the market share of each seller is 50% again. As it was before they have started moving at all, the customers however, are now required to walk a maximum distance of 500 meters rather than 250. This is a classic case of a free market leading to a situation that is sub-optimal and harming to consumers.

Why did it happen?

  1. The regulator is not strict, remember our lifeguard?
  2. Customers are willing to make the distance! If they were not willing to go more than 100 M for an ice cream cone this would never have happened.

Now let’s say there are more people are coming to the beach … Where do you think they will settle? It is clear that the most desirable position is next to the ice-cream vendors. Why should you go 500 meters whenever want an ice cream if you can be closer? Who’d want to walk more than 50 meters for it?
That’s right, they will position themselves closer to the center.

Now there are already enough customers to allow for popcorn vendors to come in. When they inevitably do they will also position themselves … in the center of course, next to the consumers.

You got the idea, this is simple, basic spatial economics.

That’s why shoe stores huddle next to one another, that’s the reason the city attracts traffic from its surrounding areas. You experience it daily, you have seen entertainment centers trying to squeeze into the same square footage while everything else around it stays vacant. The mayors of small suburb towns often complain that “the big city businesses are killing us” precisely because of that, and are looking to blame their city planners. But the truth is that these are simply the market forces at play. Even strong regulation can not cope with the attraction of the big city.

“I live in Tel Aviv because of the museums, entertainment, shops …” says the young man who came from Hadera.
“I work in Tel Aviv because here I can find the highest salaries,” says the software engineer who was born in Be’er Sheva.
“We opened the office in Tel Aviv because here is where our biggest customers are located,” says the director of the advertising agency.

All of that because customers are willing to make the distance?
Does it have anything to do with urban transportation? With traffic jams and congestion?

Yep, you bet it does.

I’ll elaborate on that model further, when you test it and draft the graphs that it creates you can start and understand city phenomena that sometimes seems to be counter-intuitive. Disappearing traffic, “leapfrogging” development, city center rise and decay, we can explain and hopefully predict parts of the “behavior” of the city.


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