Bird's-eye View Aerial City Landscape Photography

Theorists have suggested many possible reasons for why people would have originally decided to come together to form dense populations. In his book City Economics, Brendan O'Flaherty asserts "Cities could persist—as they have for thousands of years—only if their advantages offset the disadvantages" (O'Flaherty 2005, p. 12). O'Flaherty illustrates two similar attracting advantages known as increasing returns to scale and economies of scale, which are concepts normally associated with firms. Their applications are seen in more basic economic systems as well. Increasing returns to scale occurs when "doubling all inputs more than doubles the output an activity has economies of scale if doubling output less than doubles cost" (O'Flaherty 2005, pp. 572–573). To offer an example of these concepts, O'Flaherty makes use of "one of the oldest reasons why cities were built: military protection" (O'Flaherty 2005, p. 13). In this example, the inputs are anything that would be used for protection (e.g.: a wall) and the output is the area protected and everything of value contained in it. O'Flaherty then asks that we suppose that the area to be protected is square and each hectare inside it has the same value of protection. The advantage is expressed as: (O'Flaherty 2005, p. 13)
(1) O = s2, where O is the output (area protected) and s stands for the length of a side. This equation shows that output is proportional to the square of the length of a side.
The inputs depend on the length of the perimeter:
(2) I = 4s, where I stands for the quantity of inputs. This equation shows that the perimeter is proportional to the length of a side.