The City-Country Rule : An Extension of The Rank-Size Rule Rein Taagepera Edgar Kaskla

This study introduces a “city-country rule” to complement the well-known rank-size rule for cities, from which it is derived. Th e city-country rule enables us to make a rough estimate of the population of the largest cities when the population of the entire country is known. It quickly tells us whether the actual city populations are large or small, compared to the world average for similarly ranked cities in countries of comparable size. Th e existing rank-size rule describes the empirical relationship between a city’s or town’s population and its ranking relative to other cities within an interacting geographical area. Th is regularity was fi rst noted by Auerbach (1913) and later popularized by Stewart (1947) and Zipf (1949). Hence it is often called Zipf ’s law. If R is the rank in size of a given city and PR is its population, the simplest form of the rank-size rule is

rank-size rule for cities, from which it is derived.Th e city-country rule enables us to make a rough estimate of the population of the largest cities when the population of the entire country is known.It quickly tells us whether the actual city populations are large or small, compared to the world average for similarly ranked cities in countries of comparable size.
Th e existing rank-size rule describes the empirical relationship between a city' s or town' s population and its ranking relative to other cities within an interacting geographical area.Th is regularity was fi rst noted by Auerbach (1913) and later popularized by Stewart (1947) and Zipf (1949).Hence it is often called Zipf ' s law.If R is the rank in size of a given city and P R is its population, the simplest form of the rank-size rule is R P R = constant = P l (1) where P l is the population of the largest city.When graphed on doubly logarithmic paper, P R versus R, this equation corresponds to a straight line with slope -1.For many countries (or other interacting regions) a different slope (-n) yields a better data fit, and the corresponding generalized expression is Rⁿ P R = constant = P 1 .(2) As an empirical observation, the rule extends far beyond city sizes.It has been claimed to apply to the frequency of English words, the frequency of cita-A "city-country rule" derived from the wellknown rank-size rule for cities correctly predicts the average relationship between the population of a country' s Rth-ranking city (P R ) and the total population (P) of that country: P = RP R 1n(RP R ).
Th e formula applies in principle to other systems of defi ned total size, such as revenues of fi rms within industries or citations of journals within a scholarly discipline.Th e important result is that we can thus predict the populations of all cities in a country, once its total population is given, and quickly tell whether the actual city populations are large or small, compared to the world average for similarly ranked cities in countries of comparable size.For countries of more than 100 million, a more elaborate formula holds.Th e largest city in the country needs a correction (primacy factor), which vanishes as the country population increases.Actual primacy may be relative (compared to other cities) or absolute (compared to country population).

abstract
Rein Taagepera & Edgar Kaskla 158 The City-Country Rule 159 tion of scholarly journals, the size distribution of organisms in an ecosystem, the revenues of fi rms in an industry, and even the magnitude of earthquakes.We prefer to call it a "rule" rather than "law," because no widely accepted theoretical foundation has been supplied, thus it remains an empirical rule rather than a scientifi c law.
An extensive literature has attempted to explain the existence of such a ranksize relationship.Berry (1966) in particular has drawn upon the earlier work of Simon (1955) to suggest that the rank-size regularity is based on purely mathematical grounds through stochastic growth models.But other works (Vining 1974, Okabe 1977) have challenged the mathematical assumptions that produce such a "steady state" model.Further approaches (Dacey 1966, Beckmann and McPherson 1970, Mulligan 1976, Beguin 1979) have focused on the assumptions that produce a continuous rank-size relationship rather than a stepped or hierarchical relationship that would be consistent with central place theory as developed by Christaller (1966). 1mpirical testing has also continued on the level of individual countries and regions past and present (Berry and Kasarda 1977, Skinner 1977, Alden 1979, Asami 1986, Vining 1986) as well as for the world as a whole (Chase-Dunn 1985, Ettlinger andArcher 1987).Agreements and disagreements with the rank-size rule have been observed, as well as diffi culties in testing it, since "legal" (administrative) and actual sizes of population centers diff er (Asami 1986) and national units may not always be the most suitable units in which the test should be carried out.
Further operational diffi culties have been noted in defi ning (1) the limits of a city system and (2) the precise meaning of "primacy" (Walters 1985).Th e latter term expresses the observation that the rule at times works with all other cities except the largest one, which tends to be larger than expected.Th is "primacy factor" will be discussed in more below.Now we come to the new idea investigated in this study.Th e rank-size rule has been a rule about sizes of components (such as cities).But whenever these components add up to a defi nable total, it also has implications regarding this total.Cities and other smaller settlements do add up to the total population of the country (or other territorial unit).2Our purpose here is to make this implicit relationship explicit and to use it in predicting population sizes of cities in a given state system.By so doing, we will expand the scope of the rank-size rule considerably.Th is rule is all too often dismissed as being an interesting empirical regularity without much applicability as a predictive tool.Th e new "city-country rule" does predict the sizes of all cities in an average system, on the basis of total population.Deviations from the predictions indicate that the actual system is diff erent from this average.Primacy, too, can be defi ned and measured in a more precise way when the total population of the country is taken as the base line.

the model for the city-country rule
Assume that the rank-size rule (Eq.2) applies to all settlements, down to the last isolated one-person dwelling. 3By summing up all these populations, one should get the population of the entire country.Th e details are worked out in Appendix A. If n=l (Eq.1), the following equation expresses the population P of the entire country (or other suitable territorial unit) in terms of the population P R of its R-th largest city or settlement: For the largest city in the country (R=1), this becomes P = P 1 1n P l , (4) and for the next-largest city (R=2), it is P = 2 P 2 1n(2 P 2 ).( 5) The general equation (corresponding to Eq. 2, with any slope indicator n) is more complex: The City-Country Rule 161 P = R P R (P R -l+l/n -R n-l ) / (l-n).( 6) In the case of the largest city (R=l), this yields P = P 1 (P 1 -1+1/n -1) / (1-n), ( 7) and for the next-largest city (R=2), it becomes P = 2P 2 (P 2 -1+l/n -2 n-1 ) / (1-n).(8) We will test the city country rule with worldwide data for the largest and second-largest cities (Equations 4, 5, 7 and 8), as well as for the third-, fourthand fi fth-ranking cities (using the analogous equations derived from Eqs. 3 and 6).Th e population of the largest city often deviates from the rank-size rule, which sets in with the next-largest city.Th erefore, the relation between the populations of a country and its second-largest city (and the third-, fourth-and fi fthlargest) is expected to give a better fi t than is the case with the largest city.We will start the test with these next-largest cities.Only if this fi rst test turns out fairly successful would there be any point in testing for the largest cities, so as to evaluate the extent of the primacy factor.

testing the city-country rule
In order to test the implications of the rank-size rule for the total population of countries, we collected data for 203 formally independent or eff ectively autonomous countries as given in Th e World in Figures (1988).Th e data refer to the period around 1985 and include the populations of countries and up to fi ve largest cities, with increasing gaps for the lower ranking cities, especially for countries of less than half a million people.For the fi fth-largest cities the number of cases decreases to 124.
Before proceeding to the direct test, we should determine the median value of power index n (assuming that countries do follow the rank-size rule).Th is is done in the following way.Figure 1 shows the population ratio P 4 /P 2 of the fourth-and second-largest cities graphed against the population of the entire country. 4According to Eq. 2, the median ratio should be ½ n .While the points are extremely scattered, the median values of P 4 /P 2 (indicated as x-es in Figure 1) are strikingly close to ½, at any country population below 300 million.Th is suggests that, on the average, the same value of n applies to countries of almost any size, and this value is n=1.0, in line with the simplest form of the rank-size rule (Eq.l).For the world as a whole (as also shown in Figure 1) a lower value of n=0.85 would apply, and the same tends to be the case for continents.
In this light, what could be considered a successful test of the city-country model?If we graphed the population P 2 of the second-largest city versus the country population P, all points should fall reasonably close to the curve corresponding to n=l.0 (as given by Eq. 5).In order to be consistent with the fi ndings of Figure 1, we would wish for most data points to fall in the narrow zone delineated by the curves for n=0.9 and n=l.l (as determined by using Eq.8).Moreover, this should be especially the case for country populations below 300 million.For the larger countries, a deviation toward lower values of n is expected in the light of Figure 1.
Figure 2 shows P 2 (the size of the second-ranking city) graphed against P (the population of the country), both on logarithmic scales.Th e theoretical curves for n=0.9, 1.0 and 1.1 (based on Eq. 5 and 8) are shown. 5Th e actual data 4. Why pick the fourth-and second-largest cities?We want to avoid the largest cities, where the primacy factor enters.Beyond the fourth-largest cities, gaps in our data set develop.Within this range we want to have suffi cient contrast; hence P4/P2 is chosen.Countries at exactly 10 million-Belgium, Cameroon, Cuba, Greece, Ivory Coast and Madagascar-are labeled in Figure 1, because we will use them in a later section.Crosses indicate median values for the given population range.

5.
Th ese curves are not straight lines on the log-log graph, but the deviation from straight lines is almost invisible throughout the range shown.For reference, the line P2=P/2 is also shown.Th is is the maximum conceivable size the second-ranking city could have; it would be reached when a country consists of just two cities with equal populations.

Rein Taagepera & Edgar Kaskla 162
The City-Country Rule 163 points are visibly most crowded in the zone between the n=0.9 and n=l.l curves, and n=1.0 is close to the best fi t curve for these data.Countries with very large populations (India and China) are on the low side, as expected.Th us our simple model does agree, indeed, with the pattern observed.
A similar test was carried out for the third-, fourth-and fi fth-largest cities. Th e patterns of the corresponding graphs (not shown here) are quite similar to that in Figure 2 in that most country points fall between n=0.9 and n=1.1.As shown at the bottom of Table 1, more than 50% of the data points for the second-largest cities are located within the zone between n=0.9 and n=l.l.For the next-largest cities, this proportion gradually increases to 64%.
Th e main body of Table 1 lists the median values of power constants (n) at various ranks (from R=1 to R=5).Th is is done separately in various country population brackets.Th e median n for all cities in all countries is 0.99.Th e fi rstranking cities will be discussed later.For all lower-ranking cities the values of n are within plus or minus 0.07 of n=1.00, as long as the total populations of countries remain below 100 million.For countries larger than 100 million, the values of n are consistently below 1.0, with a mean of 0.94, and for the world as a whole the values of n drop to around 0.85.Th e median n for all second-ranking cities is 0.98, and for the next-ranking cities it slowly but consistently decreases to 0.96 for the fi fth-ranking cities.
Th us, the average relationship between the populations of a country and of its second-to fi fth-largest cities is well predicted by our model, despite its very simple starting point.Except for very large countries, the value of the parameter n that yields the best overall fi t (.99) is extremely close to 1.00.
We may well wonder whether the simple relationship with n=l is some theoretical norm for city systems which are genuinely separate interacting systems rather than parts of a larger system or composites of several systems.In this light, the lower n for very large countries could mean that they consist of several separate geographical systems.If so, then n should be especially low if the entire world were tentatively considered a single system.Th is is, indeed, the case (see Table 1): for the world, n is around 0.85.Going in the reverse direction, the tiniest countries still do not have n appreciably larger than 1.0, suggesting that, on the average, they are separate systems rather than parts of larger regional systems.

the primacy factor
It is now time to apply the model to the fi rst-ranking cities.Because of the primacy factor, we expect a shift away from the n=l.0 curve.As a test of these expectations, we will again use the format of Figure 1. Figure 3 shows the population ratio P 2 /P l of the largest and the second-largest cities graphed against the population of the entire country.(Th e island countries, marked with a special symbol, are discussed in Note 6.)If there were no primacy eff ect, we should have the same median n as in Figure 1 (n=1.0),and hence a median P 2 /P l =0.5.Th is is visibly not the case.Th e median values of P 2 /P l at various P (indicated as x-es in Figure 3) are clearly below the 0.5 level, except for country populations above 100 million.
What this means is that for country populations of less than 100 million there is a primacy eff ect.It tends to be most marked when country population remains below 10 million.For country populations of more than 100 million the primacy eff ect vanishes or even reverses itself, in a pattern similar to that observed in Figure 1.
Th e primacy question has often arisen, also in relation to theories of underdevelopment and dependency (see Ettlinger and Archer 1985).Our test suggests that primacy is largely the consequence of being a small country.Within the present context of transnational economic interdependence (whatever its causes) and communication linkages, the largest city in a small country must be in close contact with other large cities in a wider region of the world.Th ese strong connections to centers of other countries present economic opportunities that provide their own impetus for further development of the largest city in a small country, leading to a hierarchy of central places that follows the pattern suggested by Christaller (1966) and Lösch (1954).All other things being equal, the development of central places must be analyzed in terms of the location vis-a-vis other central places, and so too the ability of a threshold population to support the variety of activities found in the highest-ranked centers.Political borders play an important role in small states, because they artifi cially limit the area to be serviced.Within that area the capital city may turn out to have a greater variety of services than the major city of an equally large area within a larger country. 6 6. -and archipelagoes may have a weakened primacy eff ect.

Rein Taagepera & Edgar Kaskla 166
The City-Country Rule 167 We are now ready to test the city-country model for the largest city in the country, using the format of previous Figure 2. We expect the likelihood of primacy to decline as total population increases and a greater number of other relatively large and hence multifunctional cities is introduced.Figures 1 and 3 confi rm this expectation: the ratios P 2 /P l and also P 4 /P 2 increase as total population becomes very large.Figure 4 shows P l graphed against P. Th e maximum conceivable population the largest city could have is P 1 =P, when the country consists of a single city.Th e theoretical curves in Figure 4 are based on Eq. 4 and 7. Countries of about 100 million are the only ones for which we would expect the data points to fall mainly in the zone delineated by the curves n=0.9 and n=1.1.For larger countries many points are expected to fall below that zone, as in Figure 2.For smaller countries, many points are expected to be above that zone, because of the primacy eff ect.Th e pattern of actual data points is fully as expected.
Th e previous Table 1 shows the median values of n for the largest cities, indicated as n(P 1 ), for various ranges of country population.Th ese values are close to 1.00 only for country populations over 10 million, refl ecting an absence of primacy eff ect.As countries become smaller, n(P 1 ) rises markedly above 1.0, indicating an increasingly strong primacy eff ect.For the world as a whole, n(P 1 ) falls clearly below 1.0, and its value is, in fact, lower than those of n(P 2 ) to n(P 5 ).Close to 50 percent of the largest cities still are located in the zone between n=0.9 and n=1.1, but this fi gure is somewhat misleading, since the deviant points are preponderantly above this zone rather than spread equally on both sides.

prediction and evaluation of city populations
Th is is the payoff section.Th e city-country rule allows us to predict the world average populations of cities for a given country population.It then enables us to say whether cities in a given country are relatively large or small, compared to this world average.Th is is something the rank-size rule cannot do.
In terms of prediction, the best that rank-size rule can off er is to take the size of one city (maybe that of the second-largest, to avoid the primacy eff ect) and predict the rest.Th is is a weak peg on which to hang the whole scale, because the given city could be relatively small or large compared to all others.To guard against that, one would have to graph P R against R and then look for a city that fi ts the average pattern-but that changes prediction into a retroactive search for a systematic relationship.In contrast, the population of the entire country is a much more stable comparison point, since it includes the populations of all settlements.Once the country population is given, the city populations are bound to fall in place to some extent.
According to Table 1, the simplest form of the rank-size rule (that with n=1.0) applies on the average over a wide range of country populations (10,000 to at least 100 million).If we know the country population P, we could use P=P 1 1nP l (Eq.4) to determine a theoretical value of P 1 , and then use RP R =P 1 (Eq. 1) to calculate all other city populations.Two technical problems arise: 1) in the equational form above, P could be directly calculated from P 1 but not vice versa; and 2) the value of P 1 obtained does not include the primacy eff ect.Th ese issues are addressed in Appendix B. Both axes use logarithmic scales.

Rein Taagepera & Edgar Kaskla 168
The City-Country Rule 169 Figure 5 shows the resulting average expectation and actual data for six median-range countries with the same population: 10 million (plus or minus 0.20 million, according to Th e World in Figures, 1988).Both axes again use logarithmic scales.Th e dashed curves indicate a deviation from expectation by a factor of two.Th e median pattern of the six countries is somewhat steeper than predicted.Th e median P 1 (1.4 million) is very slightly on the high side.Th e median P 2 (0.4 million) is exactly as predicted.Th e median P 3 to P 5 are low by a factor of two. 7 How do the individual countries fare?Belgium fi ts the expected average pattern very closely, and so does Cuba, except for an unusually large capital city.In Greece the two largest cities are unexpectedly large, while the lower-ranking cities are small.
If one used simply the rank-size rule, Ivory Coast would seem to satisfy it (apart from a strong primacy factor), since the points for second-to fi fth-largest cities fall on a straight line (with a very steep slope corresponding to n=1.9).From the perspective of the city-country rule, however, all cities beyond the fi rst are much too low.In Madagascar, the rank-size rule would lead to the impression that heavy primacy depresses the population of all other cities.Th e citycountry rule adds the extra insight that even the largest city itself is rather small for a country of that population.In Cameroon, too, what appears a two-city primacy (akin to Greece) turns into two-city normalcy, plus unusually small thirdto fi fth-ranking cities. Th us we have to specify two types of primacy: relative primacy (compared to other cities) and absolute primacy (compared to the country as a whole).
conclusion Th e model presented pegs the entire rank-size pattern of cities to the total population of the given country.It thus leads to new ways to make worldwide comparisons, expanding the scope of the rank-size rule appreciably.Th e test with second-ranking cities in Figure 2 (and also with the third-to fi fth-ranking cities) is successful in confi rming the empirical validity of the simplest formulation of the rank-size rule (n=1.0),since the actual data points essentially scatter evenly around the n=1.0 curve at all population sizes.Th e city-country rule also leads to the development of some ways to deal quantitatively with the question of primacy.Th e test with fi rst-ranking cities (Figure 4) agrees with the general model qualitatively.Although we still lack a quantitative model of primacy, at least we obtain an empirical expression for the worldwide average primacy factor as a decreasing function of the country population. 8 On the basis of this information, it is now possible to develop an expected city population profi le for individual countries of a given population, as illustrated in Figure 5.We can tell not only how cities stand compared to other cities in the same country (as the rank-size rule does) but also how any of these cities stand in comparison with similarly ranked cities in other countries of similar population.In particular, primacy can now be measured not only in comparison with other cities in the same country but also in a world context.
How satisfactory is the fi t in Figure 5? Is it suffi cient that Belgium, Cuba and (marginally) Greece agree with the expectations only within a factor of 2? Is the model failing in the case Cameroon, Ivory Coast and Madagascar where the deviation from expectation is even larger?It depends on the criteria for satisfactory.Th ere is the absolute one: How close are the data to an ideal fi t with the model?But there is also the relative one: Is the model doing better than competing models?In the latter respect, the answer is a defi nite "yes," because to our best knowledge no other models have been presented to tie city populations to the population of the entire country.In this sense, we have made considerable progress. 97. Th is subset of countries considered covers the typical range of the ratio P4/P2 in Figure 1, but the median ratio of the subset (0.35) is below the general value of 0.5.Th e median ratio P2/P1 (0.21) is also on the low side compared to the average of countries of about that size in Figure 3, which is 0.33.
8. Scholars used to dealing with empirical data fi ts may be concerned that the R-squares in Figures 2 and 4 are only around 0.8, while they are typically higher than 0.9 for rank-size fi ts.If so, what have we gained?Th e rank-size fi ts usually apply to a single country, while here we are dealing with all countries in the world.Th e surprise actually is that worldwide scatter is only slightly less than for individual countries.
9. Th e strong deviance from expectation in the case of the African countries in Figure 5 leads to a broader question.In the case of low urbanization all cities in a country of a given population are almost bound to be smaller.If so, can the model be expected to hold in earlier historical periods when urbanization worldwide was appreciably lower?Of course, organized countries in earlier times also tended to have smaller total populations (Taagepera 1997), but the question remains.

Figure 1 -
Figure 1 -Population ratio of fourth-and second-largest cities as a function of the total population of the country.

Figure 2 -
Figure 2 -Population of the second-largest city as a function of the total population of the country.

Figure 3 -
Figure 3 -Population ratio of second-largest and largest cities as a function of the total population of the country.

Figure 4 -
Figure 4 -Population of the largest city as a function of the total population of the country.

Figure 5 -
Figure 5 -City populations in six countries with exactly 10 million population: actual and as predicted by city-country rule with primacy adjustment.