In the paper, the authors develop a model that links land rents, land consumption (for the rearing of children) and fertility in urban areas, and they particularly distinguish between fertility in the urban core and the suburbs (or urban fringe). The main argument in the paper is that families need land to raise children. Land is more expensive in the urban core than in the suburbs, so when people want children they move out of the central city. However, land is expensive to rent and so rents also have an effect on the number of children that are raised. In the theoretical model Aiura and Sato develop:
...the land rent is higher in the central part of the city, leading to lower land consumption and fewer children. Moreover, city growth results in increases in land rent, which in turn results in a decline in land consumption and fertility.Most of the paper is quite mathematically technical. However, the authors then go on to calibrate their model with data from the Tokyo metropolitan area over the period from 1950 to 2010. For the most part, their model does replicate the observed results for Tokyo, although the observed fertility differential between urban core and urban fringe is smaller than the differential from the model. However, the key results still hold: the total fertility rate is higher in the suburbs than in the urban core, and as the city grew, total fertility fell.
What should we take away from this? I don't think it explains all, or necessarily even a large part, of the decline in total fertility that countries have been experiencing. However, it may help to explain some of the localised differences. I wouldn't be at all surprised if total fertility was lower in central Auckland than outside the urban core (though there are ethnic differences at play across those areas as well).
Also, we currently live in a period where house prices have been rising (particularly in the large urban centres like Auckland). If this research holds for New Zealand (as it appears to for Japan), we might expect further declines in urban fertility as a result, and lower than expected fertility to persist as long as house prices remain high.