RDP 2001-08: City Sizes, Housing Costs, and Wealth 5. A Two-factor Model of City Populations

As discussed by Krugman (1996), models generating power laws for their size distributions generally involve the interplay of a centripedal force encouraging population into large agglomerations, balanced by a centrifugal force such as congestion costs, that limits this tendency to agglomeration. In this section, we outline a simplified model that generates Zipf's Law for some parameter values, primate cities for others, and flat rank-size relationships similar to that in Australia for others.^[20] Although this model clearly excludes some important details of the evolution and growth of cities, it captures some essential features that may generate insights about the forces driving both the urban structure and relative housing costs seen in Australia.

The household location decision involves a trade-off in which households compare the employment benefits that large cities offer, against the increased costs of congestion, proxied by high dwelling prices. In our model, two types of firms demand labour: local and national firms. Local firms sell only into the city market they are in, and compete only with other firms in that city. The number of such firms is random but roughly proportional to city populations. National firms on the other hand sell into the entire national market and locate so as to minimise transport and land costs. This creates a tendency for national firms to locate in the largest city and fosters the formation of large agglomerations, although this is partly offset by land costs which we proxy by housing costs. However, the size of cities is constrained by housing costs, which rise with population size. Since households prefer to minimise commuting times, they are willing to pay a premium to live close to the city centre. This tends to raise housing prices as the city grows, discouraging the formation of large agglomerations.

The relative importance of these two effects depends upon two key parameters: the share of national firms in the economy (β) and transport costs (θ). Our model assumes the birth rate in city i is a random variable that has a common variance across cities and a mean that is scaled by attractiveness of that city relative to the national average of all cities. For each pair of values for the parameters β and θ, we conduct 500 simulations, each of 500 periods for a country of 100 cities. When the share of national firms is small, we can generate rank-size relationships consistent with Zipf's Law. However, as β and θ rise, the Zipf curve tends to flatten as the largest city commands an increasing share of the population and the national average house price rises. This distinguishes our model from previous random growth models which could only generate city size distributions consistent with Zipf's Law, and not deviations from it. Moreover, our model of city formation captures the Australian experience: countries with relatively small populations (high β) spread over large distances (high θ) will have more concentrated populations and higher average housing costs than countries without these characteristics.

Footnote

The details of the model are provided in Appendix A. [20]