Territorial Division: The Least-Time Constraint Behind the Formation of Subnational Boundaries Abstract: Nations usually locate their smaller administrative subdivisions in re- gions of highest population density. This report derives a precise form of the size- density relationship from the general assumption that social structures evolve in such a way as to minimize the total time expended by society in their operation. The result is confirmed empirically.
All modern societies are subdivided in- to sets of primary political divisions (for example, states, counties, department). Where societies exhibit internal variation in population density, the smaller territo- rial units tend to be located in the more densely settled region (1). This negative relationship between size and density can be derived from the general assumption that social structures evolve under the constraint of minimizing the total societal time expended in their operation.     Territorial subdivision results from the necessity for people to travel between dispersed residences and some central place (for example, a county seat) under limiting conditions of time (the 24-hour day) and time-saving technology (the av- erae velocity of the means of trans- portation). If territorial divisions are too large, portions of the population will not be able to interact with a center. If divi- sions are too small, the cost of maintain- ing the centers would be unnecessarily high, assuming there were enough local resources to maintain them at all. The theoretical derivation will develop equivalencies between these opposing cost factors and societal time expendi- tures, determine the condition under which total time expenditure would be a minimum, and show that the negative size-density relationship follows from this condition.     Imagine an undifferentiated plane which is to be divided into territorial units, each containing a cener designed to serve the population associated with it. Select an imaginary unit and call its area A and its population P. Now let S repre- sent the average travel distance to the center, given the distribution of the popu- lation within the unit. This average dis- tance, divided by the velocity of the means of transportation v, gives us the average travel time expended by the pop- ulation in using the center.     Maintenance of the center and provi- sion of its services to the population will require a further time expenditure, both in direct man-hours of work and in the form of indirect costs paid by the popu- lation to support such work. If we let h represent the time cost of maintaining the center, divide this cost by the total popu-		lation, and add the result to the term S/v, we obtain the expression T = S/v + h/P (1) where T is the average societal time ex- pended in using and maintaining the serv- ice center of the territorial unit.     Since our task is to find the area which will minimize average time expenditure, we must introduce A in both right-hand terms of Eq. 1. Simple dimensional analy- sis (2) suggests that the average distance S will be proportional to the square root of the area A, regardless of the shape of the terriorial unit. Thus, with g as the constant of proportionality, we have the substitution S = g√A for the first term. The constant has ben evaluated for cer- tain regular plygons which occur fre- quenctly in the study of spatial relation- ships (3); its exact value will not be essen- tial in the present derivation. The definition of density, D, as ppopulation per unit area (D = P/A) permits subsitution of AD for P in the second term of Eq. 1 to yield T = g√A/v + h/AD (2) from which we obtain the derivative dT/dA = g/(2vh√A) + h/A2D (3) which, set equal to zero and solve for A, gives us A = (2vh/gD)2/3 (4) as the condition under which T will be a minimum (the second derivative of Eq. 2 can be shown to be greater than zero).     Holding v, h, and g constant, we can obtain a linear form of Eq. 4, relating areal size to density, log A = K - 2/3(log D) (5) where K is the log of 2vh/g to the two- thirds power. Equation 5 readily lends it- self to emprirical test with a least-squares estimator to deterine the slope relation log-size to log-density.     Such an analysis has been carried out for 98 modern nations (4). While the slopes for individual nations vary some- what around the expected -2/3 value (and in some cases the number of subdivi-
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divisions within a nation was too small to per- mit adeuate statstical test), the aggre- gated 1764 political subdivisions did yield a regression slope between log-size and log-density of -0.66, a result which very clearly conforms to the theoretical expec- tation developed here. It would thus seem that the general sssumption, that social structures revolve in such a way as to min- imize the societal time which must be ex- pended for their operation, is sufficient to account for the observed empirical rela- tion between size and density. G. EDWARD STEPHAN Department of Sociology, Western Washington State College, Bellingham 98225 References and Notes 1.    P. Haggett, Locational Analysis in Human Geog- raphy (Arnold, London, 1965); W. Skinner, J. Asian Stud. 24, 3 (1964); E. Stephan, Am. Sociol. Rev. 36,451 (1971);_____ and S. Wright, Ann. Regional Sci. 7, 113 (1973); E. Stephan and L. Tedrow, Pac. Sociol. Rev. 17, 365 (1974); S. Webb, Sociol. Soc. Res. 58, 387 (1974); E. Ste- phan, Am. Sociol. Rev. 41, 569 (1976). 2.    W. Duncan, Physical Similarity and Dimension- al Analysis (Arnold, London, 1953).		3.    The value of g  for all n-sided regular ploygons, including the circle as a limiting case, can be de- termined as follows: let the centroid of the n-gon be the origin of the polar axis bisecting one of its central angles, forming the angle φ (= π/n) with an adjacent radius R of its circumcircle. The axis, radius, and n-gon edge will form a triangle with area equal to ½ R2cosφsinφ. The average dis- tance to the center of the n-gon will then be given by E(r)  =    2 R 2cosφsinφ      φ ∫ o   ψ ∫ o     r  2 dr dθ where E(r) is the expected value of the distance between the origin and a random point (r,θ) within the triangle, and ψ = Rcosφ/cosθ, the up- per limit of r given θ. Evaluating the integral gives E(r)  =    R  3    ( 1 +    ln (secφ + tanφ) secφ tanφ   ) which, divided by the square root of the area of the n-gon, gives the value of g. In practice, the only n-gons which attract the attention of region- al scientists are the triangle, square, hexagon, and circle; the first three permit tight packing in lattice structures and the fourth is useful in ideal constructions around a central point. The value of g is 0.4037 for the triangle, 0.3826 for the square, 0.3772 for the hexagon, and 0.3761 for the circle. 4.    E. Stephan, Am. Sociol. Rev. 37, 365 (1972). 5.    I thank Albert J. Froderberg of Western Wash- ington State College for his help and advice on this report. 13 January 1977
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