gradients across developed and developing countries suggests that (i) density functions tend to be flatter in countries with higher incomes.<sup>44</sup> For example, Japan and the Republic of Korea had similar population densities in 1970s, yet the average 1970 gradient for Korean urban areas was 0.670, whereas the average for Japanese urban areas was 0.391. (ii) Although the tendency for gradients to decline as time passes is present in both groups, nonetheless it is more pervasive and persistent in the case of developed countries. For instance, the average density gradient for Japanese urban areas declined to 0.46 in 1965 to 0.39 in 1970 while for urban areas in Korea it declined from 0.70 to 0.67 during the same period. In the case cities in Ghana, this gradient did not decline at all over the mentioned period. In fact, in developing countries, there are also some exceptions to this downward trend in density gradients, such as the smaller urban areas in Mexico and in India during the early years of the post‐World War II period.<sup>45, 46</sup> The standard monocentric model predicts that land prices fall with distance from the CBD. This has been found to be true, for instance, in the case of New York starting from 1835 when the land price gradient was ‐0.65 implying that land values declined by 65% per mile from the CBD of New York. The fit of this semi‐log standard Mill’s type monocentric urban land values model was 0.71. Contrarily, by 1900, the land price gradient had declined to ‐0.14 and the fit of the monocentric model had fallen to 0.06.<sup>47</sup> In the case of Chicago, evidence on land values reveals a similar story suggesting that the explanatory power of the standard monocentric model has declined over time.<sup>48</sup> In general examples from cities in the developed world suggest that the shape of cities evolves over time where a monocentric shape is replaced over time by a polycentric form with multiple sub centers. In the case of developing countries, this transition could be slower or even perverse as observed in the case of Moscow, Brasilia and Johannesburg.<sup>49</sup> Although the monocentric model is very useful, it is difficult to apply it to suburban regions of cities such as New York, Chicago and San Francisco with multiple nodes of employment and where distance from CBD does not monotonically determine employment density and land values.<sup>50</sup>

> 44 See Mills and Tan (1980) for earlier evidence presented here and also see Bertaud and Malpezzi (2014) for more recent analysis. The latter study suggests that cities such Shanghai, Guangzhou, Singapore, Toulouse show the steepest slope (negative) of the density gradient while at the other extreme are cities such as Mumbai, Johannesburg, St. Petersburg and Moscow that show a positive gradient.

> 45 Spatial development of African cities in Ghana also witnessed a similar evolution of urban form as those in other developing countries (see Asabere and Owusu‐Banahene, 1983). Work on 5 cities in Ghana reveals that although the average density gradients in these cities are relatively higher than those found for western and Latin American cities, they are nonetheless lower than those found for Korean and Indian cities.

46 Clark (1951) was perhaps the first to show negative association between population density and distance from CBD. However, a high R‐square in such models is not necessarily supportive of the monocentric model because it results mainly due to the inherent smoothing associated with the ring approach. In fact, the R‐square associated with the regression of density in small areas within a city, such as tracts, on their distance to the CBD is much lower because there indeed are areas of fairly high density that are located relatively far from the main centre (see Duranton and Puga, 2014, 2015) 47 Atack and Margo (1998) 48 Mills (1969) 49 Bertaud and Malpezzi (2003, 2014) 50 Duranton and Puga (2015) describe that the monocentric city model explains patterns of residential land use and commuting within a city but not why individuals wish to be in a city to begin with. Neither does this model explain the complex pattern of division of land between residential and commercial uses within a city. Nonetheless, Bertaud and Malpezzi (2003) argue that the negative‐exponential parameterization of the monocentric model can hold so long as there is “a higher density of employment at the center than elsewhere”.

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