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POSITION STATEMENT
The description of our meeting identifies three issues. The first is externalities, the second is increasing returns, and the third is spatial econometrics. For my contribution to our conference preparation, I'd like to call attention to a topic that involves all three issues.
The topic is the spatial concentration of crime, which Scott Freeman, Jeff Grogger and I wrote about in a paper in the JUE. In that paper, we argue that crime involves a spatial externality that leads to increasing returns to criminal activity, and thus to the agglomeration of that activity. Though this is not part of the paper, I also think the model leads to some difficult issues in spatial econometrics.
Here's the model. Crime is an economic activity. Individuals compare the expected return in that activity with the return in legal activities. They choose the activity with the highest expected return. The return to crime is determined by the rewards from a successful crime (the take) and also by the probability of being arrested. The higher is the probability of being arrested, the lower is the expected return. The probability of being arrested is determined by the police force in an area, but it is also determined by the number of other criminals. The larger is the number of criminals, the smaller is the probability that any one will be arrested. In that sense, any one criminal creates a positive externality for other potential criminals, an externality which causes increasing returns to scale.
In the paper, we thought about this externality operating on a neighborhood level. Because of the increasing returns, there can be two types of border equilibria for a neighborhood. It can be crime-free, a situation in which the very fact that there are few criminals means that the probability of arrest is very high and thus that the return to crime is very low. It can also be crime-ridden, a situation in which there are many criminals and the probability of arrest is very low. If the take from a crime were exogenous, this equilibrium would have everyone involved in crime. We ruled this out by assuming that the take was decreasing in the number of criminals. Regardless of that little fudge, the basic idea is that increasing returns creates two possible equilibrium, both extremes.
Also, in this model, crime increases in a city because more neighborhoods turn from crime-ridden to crime-free, not because crime increases in every neighborhood. And, large interventions could tip a neighborhood from one equilibrium to another. For example, a massive increase in the police force might tip the scale from crime-ridden to crime-free. Or a big increase in the juvenile population in a neighborhood might have the opposite effect.
The model is extreme, but I think it does capture an essence of reality. The question is whether it stands up to empirical scrutiny. Here's where the econometrics comes in. Suppose one had data on crime rates across neighborhoods at two point of time. If we were to time-difference the data, we would expect to see a tri-modal distribution. Many neighborhoods would remain the same. Some would go from very low crime rates to very high crime rates, and some would go in the opposite direction. The question is how does one distinguish that pattern from, say, a normal distribution. One possibilty is some kind of mixture of normals.
That's one problem. The other is the definition of neighborhood. On what spatial scale is this externality operating? Conceptually, we might think of the problem this way. Start with neighborhoods defined on a very small scale. It seems to me that, at a small scale, you would be most likely to pick up the tri-modal pattern if it exists. But, you'd also get a lot of noise, making it difficult to pick up anything. As you increase the scale of neighborhoods, you reduce the noise, but you are also more likely to average over the two types of neighborhoods. Is there a way that the data itself could be used to resolve this issue?
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