For the period 1987-2019 the population in Denmark increased by 13.6% corresponding to an average yearly growth rate of 0.4%. For the first half of the period the average growth is 0.3% and for the latter part of the period it is 0.5%. Due to the population growth the number of people living in Denmark increased from around 5.1 million in 1987 to approximately 5.8 million in 2019. While the population growth rate varies across time the over all trend of that of a steadily increasing population.

Figure 1 - Population growth 1987-2019

Figure 1 - Population growth 1987-2019

The population is very unevenly distributed across space. The 5 municipalities, Copenhagen, Århus, Odense, Aalborg and Esbjerg are homes of the five largest danish cities measured by number of residents. As can be seen from Figure 2, the local populations of these municipalities make up large shares of the national population compared to the Island municipalities, such as Læsø, Langeland, Ærø, Samsø and Fanø, or to the small municipalities of Dragør and Vallensbæk located in the close proximity to Copenhagen.

Figure 2 - Time averaged population shares by municipality

Figure 2 - Time averaged population shares by municipality

Obviously, the difference in population shares across areas are related to the geographic scope of the areas. If people were randomly distributed across space large geographical areas would house large populations and small geographical areas would house small populations. Or to put it differently, large variation in population shares do not in themselves imply anything about individuals behavior, the share could simply be different due the random distribution of people across geographically uneven sized spatial units.

Naive population density

One remedy for this is to look at an areas population density. The naive density of an area is defined as \(d_{ct} := N_{ct}/A_c,\) where \(N_{ct}\) is the population of area \(c\) at time \(t\) and \(A_c\) is the time-constant geographical size of area \(c\). It then follows that the population share of area \(c\) at time \(t\) can be written as

\[(1)\ \ \ \ s_{ct} := N_{ct}/N_t = \frac{A_{c} d_{ct}}{N_t},\] such that comparing two areas \(c'\) and \(c\) on population shares involves the implicit comparison of their geographical size and their density

\[(2)\ \ \ \ \frac{s_{c't}}{s_{ct}} = \left(\frac{A_{c'}}{A_c}\right) \left( \frac{d_{c't}}{d_{ct}}\right).\]

Population growth and population shares

When the spatial units are kept constant, changes in the population share of one area \(s_{c't}/s_{c',t-1}\) compared to that of another \(s_{c't}/s_{c',t-1}\) is directly related to density

\[\frac{s_{c't}/s_{c',t-1}}{s_{ct}/s_{c,t-1}} = \frac{d_{c't}/d_{c',t-1}}{d_{ct}/d_{c,t-1}}\]

Population growth is strongest in the urban areas. The growth is highest in the two largest urban areas Copenhagen and Ã…rhus. However, the growth is also positive in urban areas such as Odense, Aalborg and the Triangle area. Areas far away from the urban areas have negative population growth, hence they are being depopulated.

Let \(N_t\) denote the population in year \(t\) and let \(N_{ct}\) denote the local population in area \(c\) at time \(t\). The local share of the population is defined as \(s_{ct} := N_{ct}/N_t\). The local population growth rate is defined as \[1+g_{ct}:= \frac{N_{ct}}{N_{c,t-1}} = \frac{N_{ct}/N_{t-1}}{N_{c,t-1}/N_{t-1}} = \frac{(N_{ct}/N_t)(N_t/N_{t-1})}{s_{c,t-1}} = \frac{s_{ct}}{s_{c,t-1}} (1+g_t),\]

where it has been used that \(1+g_t:= N_t/N_{t-1}\). This implies that \[(1) \ \ \ \ \frac{1+g_{ct}}{1+g_t} = \frac{s_{ct}}{s_{c,t-1}}:= 1+a_{ct},\] which states that a local area can only experience growth \(g_{ct}\) larger than than the national growth rate \(g_t\) if the local share of the population \(s_{ct}\) is growing \(s_{ct}/s_{c,t-1}>1\).

Figure 2 - Population growth rates by municipality

Figure 2 - Population growth rates by municipality

When some areas have a higher population growth rate than others it follows that area specific shares of the population similarly have varying growth rates. This is easy to see using formula (1) for two different areas \(c\) and \(c'\) to get

\[\frac{1+g_{c't}}{1+g_{ct}} = \frac{s_{c't}/s_{c',t-1}}{s_{ct}/s_{c,t-1}},\]

In particular the local populations of the urban areas represent increasing shares of the total population. Since some areas area being depopulated this implies that the variance of local population shares are increasing over time as is displayed in Figure 3.

Figure 3 - Standard deviation in local population shares

Figure 3 - Standard deviation in local population shares

This general trend is reflected in the local trends of both Copenhagen and Ã…rhus as shown in Figure 4.

Figure 4 - Copenhagen and Aarhus

Figure 4 - Copenhagen and Aarhus

However for Aalborg, Odense and Esbjerg there is no general trend and the variation over time in the local share of the population is much smaller.

Figure 5 - Aalborg, Odense and Esbjerg

Figure 5 - Aalborg, Odense and Esbjerg

These five urban areas are also tightly interconnected. Figure 6 display migration patterns in the period 1987-2019 bewteen the five urban areas.

Figure 6 - Migration patterns bewteen the top urban areas 1987-2019