# **------------------Box 4.1:  Urban equilibrium (from Combes et al., 2005) ---------------------**

The top section of the figure plots the benefits from agglomeration. The wage of residents increases with city population. This is represented by the urban productivity curve. Think of both axes as being on a log scale. The linearity of urban productivity curve indicates constant agglomeration elasticity, in line with empirical findings (Combes and Gobillon, 2015). The middle quadrant has a reversed vertical axis and represents the rising costs of living in a city faced by residents as population grows. It is convenient to think about this as representing a price of living index in a city. Unlike the urban productivity curve, the urban cost curve is log convex, which reflects sharply increasing costs as cities get large (Combes et al., 2017).

The first curve on the bottom quadrant represents real income in the city, which divides the wage by the cost of living. With logarithmic vertical axes, this curve is obtained graphically by summing the urban productivity curve and the urban cost curve on its reversed axis. The other curve represents the supply of population for the city at hand. This is the real income that residents ask for to reside and work in the city. This curve is upward sloping. Its slope reflects mobility costs and/or a less than fully elastic demand for the city caused by locational preferences. A flat supply curve corresponds to the limit case of perfect mobility. Note that this supply curve implicitly reflects the amenities that the city at hand offers. A city with better amenities would have a lower supply curve given that residents would be willing to forego some of their consumption (real wage) to enjoy better amenities.

The urban equilibrium is point E where the real income curve meets the population supply curve. The corresponding population Neq is such that the real wage that the city can offer (through the real income curve) is equal to the real wage that the marginal resident requires to be in the city (through the population supply curve). We can then read the equilibrium cost of living and wage by reading back on the first two quadrants.

We note that point E is not the unique equilibrium. The population supply curve and the real income curve also intersect at point F, but this is unstable. A small positive shock at F would create a positive wedge between the real income and what the next resident requires to come to the city. This resident would move in, increasing the wedge further. In turn, this would lead to more residents moving in until the stable equilibrium in E is reached. By the same reasoning, a small negative shock in F would lead to the disappearance of the city.

Importantly, equilibrium population is too large relative to point G. This point would allow residents to enjoy a higher real income. This is due to a coordination failure. Should the city be at point G, it would offer potential residents more consumption than they would require to move in as the real income curve at point G is strictly above the population supply curve. Hence, point G cannot be in equilibrium and new residents move in until the two curves intersect. At this equilibrium, the city is too large. However, no single resident has an incentive to leave the city and form a new one since the real income of a new city with a very small population is extremely low. Note that we did not call point G optimal. It is only a constrained optimum that embeds a series of inefficiencies. The next box explains this issue further and derives a full urban optimum.

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