Unruled Notebook

Where should a business set up its new commercial store – chain or single – to maximize its profit? There may be a demand for the store in a sparsely populated region but it may be wiser to locate the store in a densely populated region for more profit. Is there a definite correlation between population density and commercial facility density?

Similarly, if the government or citizen group wants to locate a public facility – temples, toilets, grocery store, fire station – where should that be? It is not for profit, but should be easily accessible by many. Is there a suggestive correlation in this situation?

The answer to the above two questions is yes.

There exists a definite correlation between population density and the density of facilities – commercial or public. Both the correlation are of the form D ~ ρ^α where D is the facility density and ρ is the population density. For commercial facilities the exponent α ~ 1.13 while for public facilities α ~ 0.69. The following empirical data for the USA and South Korea emphatically asserts such a correlation.

A recent paper [1] in PNAS by Jaegon Um et al. titled Scaling laws between population and facility densities also predicts such a correlation with identical exponent values through a proposed model based on economic mechanisms that mimic the competitive balance between the profit of the facilities and the social opportunity cost for populations.

The arguments for such unique exponents are simple and catchy.

For commercial facilities it is safe to assume profit as the goal. If so, more people come to your store (facility of one type), more the profit. So, one could expect, all other conditions remaining equal (like availability of supplies) a store located in a densely populated region to incure more profits than the one in a sparsely populated region. Obviously this is a highly unstable equilibrium for maximizing profits for all. Consequently, to gain more profit, it is logical to expect a store manager to move her store from a sparsely populated region to a densely populated region. But as the number of stores increase in a densely populated region, the profit for each store drops. Extrapolating this local shifts all over a region or country, one can expect in time all the stores, irrespective of their location, to receive almost similar profits. In other words, each store would have strategically located itself in such a place where the number of customers are almost the same for each of them. So, the correlation between the facility density and the population density should have an exponent close to unity. As the data above shows.

For public facilities the motive is different. It is not profit that determines their location but, safe to say, accessibility. The cheaper it is for people to travel and access such facilities, more they will be sought. So, their location is determined by how accessible they are to more number of people. Given a bunch of such public facilities, each unit moves from a location where the average travel distance is short (so, cost of travel is cheap) to a location where the average travelling distance for a larger population is large (so, cost of travel is high). Such a relocation shifts the travel cost or social opportunity cost, as the authors call it, for a population towards a mean value – reducing in the place where it moved close to and increasing slightly in the place from where the facility moved away from. After several such relocations, at steady state, one observes the exponent α ~ 0.69 as seen from the figure above.

Both these correlations have been predicted well by the proposed models in the PNAS paper [1]. The authors have shown such a correlation exists for facility density versus population density in the data for USA and South Korea. The exponent in their model are α ~ 1 and α ~ 2∕3 for commercial and public facilities respectively.

As one could expect not all commercial and public facility obey exactly those exponent values. For example, in the US, banks have an exponent close to 0.88 while in South Korea they have a value of 1.18. Whereas hospitals in the US have α ~ 1.13 while in South Korea 0.96. By defining a generalized objective function to account for these variations, the authors have proposed a generalized correlation in the form

where D(r) is the facility density and ρ(r) is the population density (r is a certain position). When β = 0 the correlation applies for commercial facility and when β = 1 it suits the public facility. The rest of the continuous values should correlate variations of the facilities.

Obviously, it is beyond the scope of the model to attempt a positive correlation for seemingly public facilities – like a place of worship – that are run for profit.

References

1. Um, J., Son, S., Lee, S., Jeong, H., & Kim, B. (2009). Scaling laws between population and facility densities Proceedings of the National Academy of Sciences, 106 (34), 14236-14240 DOI: 10.1073/pnas.0901898106