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For a particular application I'm trying to specify a model for the level of contact between two communities, similar to modeling migration. I have no data on contact rates between communities, so there's no parameter estimation here - I'm just trying to posit a reasonable model.

The information I have at hand for pairs of communities, $(i,j)$, are their respective population sizes ($P_{i}, P_{j}$) and the distance between them, $d_{ij}$. The classical gravity model of migration says that their rate of contact is proportional to the product of their population sizes and inversely proportional to the square of the distance between them. That is

$$ {\rm contact \ rate} = c \cdot \frac{ P_{i} P_{j} }{ d_{ij}^2 } $$

Now, one could more generally conceive of the model

$$ {\rm contact \ rate} = f \left( \frac{ P_{i} P_{j} }{ d_{ij}^2 } \right) $$

where $f$ is some pre-specified non-decreasing function. I considered this model because I wanted the contact rates to be bounded between $0$ and $1$, so I was thinking something like $f(x) = e^{-k/x}$ for some constant, $k$. Would this still be considered a gravity model? If not, what would such a model be called? Any references would be appreciated.

Also, a more general formulation is

$$ {\rm contact \ rate} = f \left( P_{i},P_{j},d_{ij}\right), $$

where $f$ is decreasing in $d_{ij}$ but increasing in both $P_i$ and $P_j$, is similar in principle to the gravity model - is there a name for this model? Again, any references are appreciated.

Macro
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1 Answers1

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I encountered something similar to the first one under the rather generic term "exponential gravity model" in Erlander and Stewart "The gravity model in transportation analysis" (google books link) also see e.g. Ottensman:1995 and Isard:2005. They all use some type of exponential deterrent function. Last year there also was the review by Anderson J.E. (2011) (pre-print PDF here) in the Annual Review of Economics which you might find helpful.

Momo
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  • Thanks momo, some of these are very helpful. In the book you linked to, I did notice models in the vein of $P_{i} P_{j} h(d_{ij})$, where $h$ is a decreasing function, but not exactly $f \left( \frac{P_i P_j}{d_{ij}} \right)$ - have you seen that one specifically in any of these references? – Macro Jan 16 '12 at 19:29
  • No problem. The above is pretty much all I know though. We used (borrowing your terminology) $f(x)=x^k$ ("power gravity model") once, but unfortunately the paper lies currently on ice. – Momo Jan 17 '12 at 17:10