Controlling concentration in city centres Evidence from Berlin 1881-1914

Kristoffer Möllera and Sevrin Waightsb

Abstract: The narrow streets of historical city centres are all too often characterised by horrendous

traffic problems and choking levels of pollution. Those who choose to reside in the core must either

pay huge rents or suffice with living in tiny apartments. At the same time, however, urban economics

advises us that these close quarters are a source of important agglomeration economies that boost

the productivity of cities and drive growth. We add to this debate not through attempting to

estimate how much concentration is needed but by examining a possible approach for altering the

level of concentration. Policymakers wishing to reduce the congestion of the city centre face the

problem of raising the relative desirability of more peripheral locations. Applying highly

disaggregated data for Berlin 1881-1914, we find empirical evidence in support of using transport

improvements as an instrument to raise the attractiveness of the periphery.

Version: February 2011

a Technical University of Darmstadt, moeller@vwl.tu-darmstadt.de

b London School of Economics, s.g.waights@lse.ac.uk

1

1. Introduction

Regional concentration is determined by positive and negative externalities. On the one hand, if

people and firms concentrate in one region positive agglomeration effects raise their utility. A more

concentrated city allows for example for a cheaper provision of public goods because the cost of a

public good is divided by a larger number of inhabitants (Stiglitz 1977). Moreover, internal or external

economies of scale lead to a higher a productivity of firms (Mills 1967; Dixit 1973). This way people

get a better access to a higher variety of goods and services. On the other hand, higher concentration

also implies negative congestion effects. For instance, an increasing population leads to a decline in

individual space consumption, assuming a fixed amount of space. Traffic jams and congested public

transport are further examples for the negative characteristics of highly concentrated cities. A city

reaches its optimal size in population (and thus concentration) when marginal benefits and marginal

growth counterbalance each other, i.e. an increment of population would then lead to a decrease in

utility (e.g. Dixit 1973).

If cities grow larger than their optimal size they will suffer from a decreasing utility due to

overconcentration: Even though a city is already on the downwards slope of the utility curve,

individuals do not have an incentive to move to an alternative location if its utility is relatively

smaller. Assuming a constant growth in population, the original city is over-concentrating since a

secondary city establishes too late under laissez-faire. To overcome this problem of

overconcentration, the relative utility of the periphery should be increased Anas (1992). This way

outward migration is supported and a secondary city can earlier be stabilised.

One way of raising relative utility at the periphery is to improve inter-regional transport between

core and periphery. The idea is to increase peripheral access to goods which are produced under

scale economies in the core. This way, people could benefit from a greater variety of goods and could

further reduce congestion costs by migrating outward. Based on the theoretical model introduced by

Helpman (1998) we empirically test if a location’s relative utility is determined by its relative access

to goods and services and by its relative congestion of space.

Our observation period covers Berlin between 1881 and 1914. During this time, the city experienced

important social and economic changes: The city had just become the capital of the 1871 founded

German Reich and the population of the greater area grew from 1.3 to 3.8 million (Statistisches Amt

der Stadt Berlin 1875-1920). This development was accompanied by an enormous economic upswing

driven by a rapid industrialisation (Elkins and Hofmeister 1988). Moreover, important infrastructure

projects had been completed, like an urban rail (S-Bahn) east-west connection in 1886 and the first

underground in 1902 (Leyden 1933). With a growing population and the significant transport

improvement, historical Berlin provides a suitable framework for our analysis.

The structure of the paper is as follows: In the next section we theoretically examine the problem of

over-congested cities. In section 3 we develop our empirical strategy and present the data with

which to test our hypothesis. Section 4 presents and interprets the estimation results and section 5

concludes with our policy recommendation and suggestions for future research.

2

2. Theory

2.1 The problem of the over-concentrated city

In his basic model, Anas (1992) compares two locations which are initially identical and provide the

potential ground for the emergence of new cities. Cities are inhabited by labourers who migrate to

the region with the highest utility. In a stable equilibrium, utility is equalized between the regions by

the migration process which is assumed to be quick and costless. The utility per labourer consists of

two components: (i) consumption (which is equal to individual production output) and (ii) private

land lot size. Production is characterized by localization externalities: Due to information exchange

the individual output increases with the number of labourers working in a city. The size of an

individual land lot also depends on the city’s population. Since land is assumed to be divided equally,

individuals’ consumption of space declines with population. This way Anas (1992) models the trade-

off between agglomeration and congestion forces that determine a city’s utility. The optimal city size

is reached if the ratio of these opposing forces is equal to the marginal rate of substitution

between land and consumption of goods:

⁄

(2-1)

where represents marginal utility of goods consumption, marginal utility of land, denotes the

land per labourer, the number of labourers and the marginal localisation benefit1.

Focusing on the dynamic urban development, it is assumed that in the beginning all labourers are

located in one city which is compared to a hypothetical alternative location throughout the analysis.

Starting from a population level smaller than the optimal city size, , total population is

assumed to grow continuously over time. In the original city agglomeration forces outweigh the

negative congestion effects and raise city’s utility with growing population. A few pioneers try to

settle at the alternative location, causing a low degree of random migration. Due to the low

population level in the alternative city utility is relatively low and thus the settlement unstable. The

migrating labourers quickly return to the original city. Eventually, the optimal population level is

reached and stronger congestion effects cause the city’s utility to decline with further growth in

population. The size of random migrants which is needed to outweigh the utility of the original city

and thus to destabilise the one-city equilibrium becomes increasingly smaller. Eventually, the original

city’s utility is so low that only a few migrants are needed to move to the second location in order for

it to offer a higher level utility than the first one. Workers then migrate from the original city to the

smaller city where utility is higher further increasing the level of utility there due to agglomeration

economies. This ‘panic-migration’ occurs until the new settlement becomes as congested as the

original city and they are of equal size and utility level. This is the stable two-city equilibrium.

The two-city equilibrium becomes the optimal solution at some point on the downwards slope of the

utility curve of the original city. However the system bifurcates only when the level of utility at the

original city has been driven down to almost zero. Thus it exists in an over-congested state because

1 is the production function where and .

3

the alternative location offers a level of utility that is simply too low for a small random migration of

labourers to stay there.

In order to establish a second city earlier, Anas (1992) assigns planners an intervening role as

coordinators; they should nurse a secondary city until it becomes stable. Further incentives are

needed to quickly raise utility in the new settlement. Clear signals could attract additional random

migration. The provision of infrastructure could be one way of nursing the emergence of a secondary

location or, in an intra-city environment, of sub centres (Anas, Arnott et al. 1998). The transport

improvement could increase the random migration and direct it on one point. Moreover, the

secondary sub centre could benefit from rising utility due to a better access to goods produced in the

bigger region.

The Anas (1992) model has shown that cities tend to over-concentrate under laissez-faire and

planners should intervene in order to establish a secondary city at an earlier stage. Improved

infrastructure could help to nurse an alternative location. Therefore we discuss an urban New

Economic Geography (NEG) model in the next section, which analyses the effects of transport

improvement on a two-city framework.

2.2 The effect of transport improvement

The New Economic Geography (NEG) literature (e.g. Krugman 1991) usually predicts an increasing

concentration of firms when inter-regional transport costs are reduced. Since manufacturing goods

can be shipped more cheaply, firms agglomerate to realise scale economies. This result contradicts

our aforementioned expectations. However, the NEG models need to be adapted to an urban

environment. Instead of modelling agricultural goods, Helpman (1998) introduces consumption of

housing services as the homogenous good. The idea is to have a good/service which is non-tradable

across regions and which implies a declining individual consumption share with growing population.

This way the cost of housing services is closely related to the private lot size in Anas (1992): A

growing population leads to a decreased supply of space and implies higher housing costs, describing

the dispersive effects. The manufacturing sector is modelled as in Krugman (1991). He assumes

brand specific economies of scale, so that under monopolistic competition a higher number of

labourers generates a greater variety of brands. The manufacturing goods are traded across regions

at a cost (inter-regional transport cost). The individual Cobb-Douglas utility function is:

, (2-2)

where represents a consumption index of differentiated manufacturing goods2 and describes the

consumption of housing services. Individual utility strongly depends on (i) the elasticity of

substitution between brands and (ii) the expenditure on housing . Assuming a low elasticity of

substitution, consumers demand brand specific goods and thus prefer a higher variety of goods. They

are attracted by a more-populated region where due to brand specific economies of scale the variety

of goods is larger. Conversely, with a high elasticity of substitution, consumer behaviour is hardly

2 ∫

, with as the available number of brands and as consumption of brand

. This implies a constant price elasticity of demand ⁄ for each brand. also describes the elasticity of substitution between brands.

4

influenced by the variety of goods. The expenditure on housing illustrates the importance of

housing. A larger gives individuals a greater incentive to migrate in order to avoid high housing

costs.

Like before, people choose location depending on the relative utility level ⁄ between

regions and . An individual’s utility at location is given by:

, (2-3)

where ⁄ describes the consumption of housing per resident and the second factor describes

spending per resident on manufacturing products over a price index of differentiated products .

The price index represents the price of manufacturing goods which are produced at location and

which are imported from at inter-regional transport cost as well as the preference for a certain

variety of brands. This preference is expressed by the elasticity of substitution between brands.

Depending on the assumed parameters and , changes in transport costs are expected to have

different effects on regional concentration. Regarding Engel’s Laws, rising incomes change the

consumption pattern and develop an interest in a greater variety of goods (Engel 1857). During our

observation period Berlin experienced a strong industrialisation associated with an increasing

quantity and variety of manufactured goods and an increasing wealth (Elkins and Hofmeister 1988).

Therefore we assume an increasing importance in varieties of goods akin to a falling elasticity of

substitution.

Beginning our analysis with a high elasticity of substitution or a high expenditure on housing such

that , there is a unique stable equilibrium where a location’s population is a function of its

relative housing stock. Hence, labourers care only about minimizing their housing cost and

maximizing their consumption of space. With high transport costs they only consume locally

manufactured goods since they do not have any preference for variety. In a two-city framework,

each city’s population eventually becomes proportional to its housing stock and reaches the same

utility level. This is illustrated by point A in Figure 2.1 (a), where denotes the relative utility and

the share of total population. Assuming an equally distributed housing stock between the two cities,

the population share is 0.5 at point A. The arrows represent the adjustment process: If a random

migration disturbs the equilibrium such that we reach a point located at the right side of A, then the

enlarged region will have to share the available housing stock with a higher number of people. Their

consumption of space at that location declines. The variety of goods is a little higher at this point.

However, since we assume a high elasticity of substitution the migrants do not benefit from an

increased consumption in manufacturing goods. Therefore, their relative utility is lower at this

location and they will move to the equilibrium point A. As long as holds, a city’s population

depends on the housing stock. Even when the transport cost decreases labourers will migrate to less-

populated locations in order to maximise their utility. Regional concentration decreases with

population.

With an increasing preference for a greater variety of manufacturing goods relative to housing, i.e.

, people’s location decisions will change based on the improvement in transportation. If this

improvement leads to lower inter-regional transport cost the trade-off of agglomeration and

dispersion forces depends on a region’s size: If a region is small the relative utility declines with

5

population because additional people only push up the house prices while the variety of goods stays

relatively constant. When a region’s population share grows the agglomeration forces outweigh the

dispersion forces and individuals are willing to pay a premium to enjoy a higher variety of goods. The

relative utility rises with population. At a very dense location the relative utility declines again with

population. The additional variety of brands does not compensate for the high housing costs. The

stable equilibria outcomes of this simulation are depicted by the points A and C in Figure 2.1 (b). The

arrows describe the adjustment process and illustrate the instability of equilibrium B. Even though

the two regions have the same housing stock, they eventually end up with different population sizes

but equalized utilities. The extra utility induced by a greater variety of brands in the bigger region is

outweighed by a lower consumption of housing services. Helpman’s (1998) simulations show that

increasing transport costs lead to more unequally sized regions.

When further reducing transport cost, there exists a critical value where sudden, catastrophic

agglomeration takes place (Tabuchi 1998). The two equilibria in Figure 2.1 (b) become unstable. Point

A shifts to the left and C to the right as people disperse again. Since industrial goods are now

available in any region without extra cost people minimise housing cost (independent of ) and

move to the less-populated region. Eventually, each region’s population becomes proportional to its

housing stock and reaches the same utility level. The unique stable equilibrium is illustrated by point

A in Figure 2.1 (a) which is identical to the situation with high elasticity of substitution.

Figure 2.1: Effects of transport cost on regional concentration

(a) high transport cost ( ), and

nil transport cost ( )

(b) intermediate transport cost ( )

Source: Helpman (1998).

In contrast to the classic core-periphery model (Krugman 1991) that predicts economic concentration

in one region when transport costs are reduced, the urban adaptations (Helpman 1998; Tabuchi

1998) predict a more dispersive outcome. Assuming a low elasticity of substitution between brands

or a low expenditure on housing, transport improvement fosters the creation of sub centres. Hence,

the models support Anas’ (1992) suggestion of using transport as an instrument for nursing a

secondary centre with well-established arguments provided by the New Economic Geography.

Based on the theoretical considerations we empirically test if a location’s relative utility is

determined by its relative access to goods and services and by its relative congestion of space. In our

analysis we concentrate on Berlin during the period between 1881 and 1914. Due to two major

infrastructure projects and in light of the strong population growth at that period, we would expect

6

the historical central business district (CBD) to experience a decline in relative utility compared to its

surroundings. In the next chapter, we develop an empirical strategy to test this hypothesis based on

the Helpman (1998) model and give an overview over the data we use.

3. Data and empirical strategy

3.1 Empirical strategy

Taking logs of the utility equation (2-2) from section 2 we get:

(3-1)

As mentioned in the theory section, in equilibrium utility is equalised at all locations that have a

positive population. However, an exogenous shock such as a transport improvement can lead to a

temporary disequilibrium, where smaller regions suddenly have better access to a variety of goods.

In this case, the difference in logged utilities between two places and is:

( ) ( ) (3-2)

Obviously it is impossible to directly compare the level of utility between places. However, we

propose two ways in which the desirability of different locations can be proxied in empirical analysis.

Firstly, the previously examined theory suggests that a utility differential would lead to growth in

population in location at the expense of a growth in location . In a rapidly growing city, it might be

enough to say that although all locations are growing, location would grow at a faster rate if it

delivered a higher level of utility than location . Thus we would expect the difference in growth rates

to hold some positive relationship with both the difference in housing consumption and the

difference in manufacturing goods consumption:

( ) ( ) (3-3)

Secondly, an alternative approach would be to invoke the theory of hedonic prices in housing

markets (Rosen 1974). Here we assume that both the congestion of space and the consumption of

manufacturing goods are capitalised in the land values at a particular location. Utility is equalised

across locations by the adjustment processes of the housing market. Therefore we would expect to

observe high land values (LV) in areas that had a less congested housing market and a higher

consumption of manufacturing goods:

( ) ( ) (3-4)

7

Hedonic pricing theory describes a situation of equilibrium, after land values have fully adjusted to all

factors influencing the attractiveness of each location. In disequilibrium, however, the land values

would not yet have fully adjusted to such differences in locations. In this case, assuming a functioning

housing market, we would expect to observe a more rapid increase in land values in locations with a

higher utility level:

( ) ( ) (3-5)

The following subsections describe the construction of the variables used in the empirical analysis of

the models specified above. The locational unit of analysis (the locations) is the individual plot level,

as determined by the detail of the land value data described in the next section.

3.2 Land values

The technician Gustav Müller (1881-1914) produced maps of land values at the plot level for Berlin

between 1881 and 1914. The historical maps were georeferenced in a GIS environment and land

values were extracted for seven time periods, for which Table 3.1 details summary statistics.

Table 3.1: Land value data summary statistics

Variable Observations Mean Std. Dev. Min Max

LV1881 13421 103.3 80.8 30 480

LV1890 23576 163.3 210.1 3 2000

LV1896 22718 226.0 254.7 5 2100

LV1900 24281 249.2 285.8 2 2120

LV1904 23274 300.7 318.9 5 2150

LV1910 24991 343.9 378.0 6 2250

LV1914 24558 340.5 380.5 14 2750

LVPOOLED 156819 257.5 309.6 2 2750 Notes:

LVPOOLED displays the summary statistics for the pooled data set of land values across all years.

The data for 1881 are more aggregated than for the other years, thus there is less variance in the figures.

The Müller maps have been used previously by Ahlfeldt and Wendland (forthcoming) in establishing

the impact of the development of the urban rail system on the land value gradient in Berlin over the

period 1890-1936. Whereas, Ahlfeldt and Wendland use only commercial land values averaged at the

block level, the data extracted for this study uses land values relating to all land uses at the full level

of disaggregation i.e. the plot level. Currently, the land values have been extracted at this level of

disaggregation only within a 1km buffer zone surrounding the rail lines U1 and U2.

Figure 3.1 displays the location of the of these land value plots on the street structure of modern

Berlin 2006. Figure 3.2 shows a comparison of land values in 1890 and 1914 within the buffer zone.

The land values have been spatially weighted by an inverse distance weight (IDW) measure with the

power of two, taking into account the 50 nearest land value plots.

8

Figure 3.1: Land value plots in Berlin extracted from the Müller maps (1881-1914)

Figure 3.2: IDW of land values 1890 and 1914

9

3.3 Population

The population data for Berlin refer to 94 historical villages and adjacent communities. They were

collected from Leyden (1933) and the Statistical Yearbook of Berlin (Statistisches Amt der Stadt Berlin

1875-1920) and were found to be consistent across sources. Assuming the population being equally

distributed within the built up area of village , we approximate statistical block 's population ( )

with respect to its proportion at the total village’s built up area :

∑ (3-6)

where represents the total built up area within block in m2 and is the total population of

village . Population density for the statistical blocks in 1890 and 1910 is illustrated in Figure 3.3

below.

Figure 3.3: Population density at the block level 1890 and 1910 (persons per m2)

10

Equation (2-3) from the previous section defines individuals’ consumption of housing at location as

equal to the location’s fixed housing stock divided by the total number of residents at that

location . We expect the magnitude of housing stock at any location to be correlated with the total

available land area of that location. Therefore, we find it reasonable to define a measure of inverse

density as a proxy for housing consumption. The locations take on the inverse density value of the

statistical block within they reside, simply calculated as the land area of statistical block divided it

population .

3.4 Access to good and services (AGS)

Transport improvements impact on the utility level at different locations by changing the quantity of

goods consumed at those locations. In NEG models, such as Helpman (1998), consumers demand

goods from distant locations because they gain utility from consuming a greater variety of goods.

They then incur transport costs both when they travel to the goods and when the goods are

transported to them. In this paper we assume that all travelling is done by consumers to the business

districts, i.e. for shopping trips. In locations with good accessibility to a variety of goods little needs to

be spent on transport costs and so total consumption is higher. When locations are more isolated, a

great share of expenditure is spent on transporting goods and so the level of consumption is lower.

Therefore a measure of accessibility to goods and services will serve as a proxy for the level of

consumption at location , or in the model.

In order to construct this index, we draw from two data sources. Firstly, is land use data extracted

from maps of Berlin for 1880 and 1910 (Aust 1986; Aust 1987). These historical maps illustrate

discrete land use blocks, each devoted to either industrial, residential, business or mixed use. In

keeping with the NEG literature we assume that each firm produces its own variety and that these

goods or services are to some extent substitutable. Therefore the number of varieties of goods and

services for each land use block is assumed to be directly proportional to the land area devoted to

business land use. A further assumption adopted is that the number of varieties is proportional to

half the total land area when the block is devoted to mixed use.

Secondly, the transport network of Berlin (Schomacker 2009; Mauruszat 2010; Straschewski 2011)

was constructed for each of the years for which we have land value data (i.e. 1881, 1890, 1896, 1900,

1904, 1910 and 1914) in order to calculate the accessibility of each location to the business land use

areas and therefore the ease by which individuals at different locations can access a variety of goods

and services. Figure 3.4 illustrates these different spatial data for a small section of Berlin.

The index was calculated at the plot level as follows: Firstly, the fastest travel time via the rail

network was calculated between each land value plot and every business and mixed land use area

. Travelling by the rail network from the land value plots to the land use blocks involves three

steps; (i) walking from plot to the nearest station, (ii) taking the shortest network route from this

station to the station that is closest to land value and (iii) walking from this station to land use

block . We assumed an average walking speed of 5km/hour and an average speed of rail network

travel of 33km/hour (Ahlfeldt 2007). Secondly, also calculated were the travel times for walking3

3 Walking distances here and in the network calculation were estimated as straight line distances between

origin and destination.

11

directly from land value plots to the land use blocks , since it is often quicker to walk directly than

to use the rail network. Finally the index for access to goods and services (AGS) was calculated for

each land value plot using either the direct walking time or the network travel time, whichever is

quicker:

∑ ( { })

, for all (3-6)

where is a decay parameter, is the area of business4 block , is the travel time using the

transport network and the direct walking time.

Figure 3.4: Land use and rail network (1910)

The decay parameter is a typical feature of accessibility indices in urban economics (e.g. Harris

1954; Ahlfeldt 2007). It describes the speed of decline in the weight of importance placed on

amenities with respect to a measure of proximity. Often this measure of proximity is simply a

distance-based measure and in these cases the value used for the decay parameter must partly be

based on the speed of travel. For example, Ahlfeldt (2007) uses a distance decay parameter of 0.5 to

represent spatial discounting of employment as accessed by the rail network and a parameter of 2 to

represent the spatial discounting for pedestrians.

However, since our measure of proximity is travel time, the speed of travel has already been

accounted for and the decay parameter only embodies the willingness of individuals to travel for

long durations. Varying the decay parameter varies the importance placed on goods and services at

distances. Therefore it plays a similar role as the elasticity of substitution parameter in the Helpman

4Or, as discussed earlier, one half of the area of mixed land use blocks.

12

(1998) model. To see this, consider that a zero decay parameter would mean that location of

residence is made unimportant since equal weighting is placed on goods located at any distance. This

is synonymous with the case of perfect complements where location is irrelevant since utility can

only be increased by visiting all locations and consuming all varieties of goods. The first two charts in

Figure 3.5 show the AGS indicator at the plot level for a very small decay parameter of 0.01. Clearly

there is not much spatial variation in this measure in either 1890 or 1910. Over this time period the

transport infrastructure was developed hugely and there was a significant decentralisation of the

location of firms, however this is seen only minimally in these first two charts. In Figure 3.6, this

decay parameter is shown to give very little discount for goods and services located at long distances.

Such a propensity to pay large travel time costs in search of greater varieties of goods and services

must clearly corresponds with a situation where of high complementary.

With a very large decay parameter then location of residence becomes ultimate. Individuals will want

to reside exactly where the most goods and services are produced. This is because individuals put

very little weight on goods and service that are located in any significant distance. This is

synonymous with the case of very high substitutability where individuals are not willing to pay the

cost of travel because consuming further varieties of goods and services will not increase their level

of utility.

In Figure 3.5, the last two charts display the AGS measure at the plot level when the decay parameter

is rather high, at 1. Figure 3.6 shows that this corresponds with a zeroing out at around 6 or 7

minutes, suggesting that individuals are not willing for any longer than this amount of time. Since

most rail network journeys are a little longer than this, the AGS shown in the last two charts of Figure

3.5 takes little account for the transport network and just shows the locations of businesses in the

1890 and 1910.

Clearly these two extreme situations are not going to be too helpful for our empirical analysis since in

neither case would the AGS measure particularly capture the effects of the evolution of the transport

infrastructure over the period. In the first case, because individuals will travel everywhere, improving

the infrastructure has no affect the relative desirability of any particular location. In the second case,

because no one will travel anywhere, improving the transport network also has no effect. The

Helpman (1998) model outlined previously suggests that relocations will follow transport

improvement under a situation of low substitutability of goods and services.

The middle two charts in Figure 3.5 display the AGS indicator when the decay parameter is 0.1. With

this parameter value the difference between the 1890 chart and 1910 capture both the evolution of

the transport infrastructure and the dispersion of business locations. Both these elements are

required to feature strongly in the variable in order to model the process described by the theoretical

literature.

Figure 3.6 shows that the with a decay parameter of 0.1 there is a significant discount after a travel

time of 20 minutes, which is an intuitively reasonable result. Further this value is not too far off other

used in the literature. For example Ahlfeldt’s (2007) parameter of 2 for pedestrians with a walking

speed of 5km/hour equates to a time decay parameter of around 0.17. Therefore, for the empirical

estimation we use the decay parameter of 0.1 to produce our index for access to goods and services.

13

Figure 3.5: Access to goods and services index 1890 and 1910 with different decay parameters

1890 1910

0.01

0.10

1.00

Figure 3.6: Discount weightings for different decay parameters (1.0, 0.1 and 0.01)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

We

igh

t

Minutes

0.01

0.1

1.0

14

3.5 The estimation equations

Following the above derivation and discussion of the variables and proxies we now re-examine the

empirical models. Equation (3-3) will be estimated as:

(

) (

) (3-7)

There are a few things to notice about this equation in comparison with the empirical model of

equation (3-3). Firstly, the variables and have been replaced with their respective proxies as

discussed in subsections 3.3 and 3.4 above. Secondly, for each of the variables the differences are

now taken for the locations with respect to the average values across all locations. This is a

simplifying assumption and we expect the same relationship should hold with differences from the

average values as with differences from each other value. Thirdly, the inverse density measure in this

equation does not have a subscript because it is the average of density across all time periods for

each location . This is an attempt to avoid endogeneity issues caused by having population on both

sides of the equation.

Equation (3-4) is estimated as:

(

) ( ) (3-8)

and equation (3-5) as:

(

) ( ) (3-9)

Here the inverse density measure can enter the equations with subscripts because population is no

longer the dependent variable. Finally, since we only want to estimate the capitalised effects of

space and manufacturing goods consumption we need to control for additional amenities which

could affect the land values. Otherwise the estimated changes in land values will be biased.

Therefore we will also estimate equations (3-8) and (3-9) using a two-stage approach. The first stage

of this approach is to run an auxiliary regression where we control locational factors as follows:

, (3-10)

where is a vector of location control variables and is the residual land values. We regress the

land values on location controls for each year we have land value data for. This way we control for

potential amenity effects for each period and their development over time. Secondly we use the

residual land values from the first stage from this regression to estimate the models in equations (3-

8) and (3-9). As the mean of the residuals converges to zero and logs have already been taken in the

15

first stage regression, the residuals of (3-10) enter directly as dependent variable in these

estimations.

In the next section we estimate these three equations (3-7), (3-8) and (3-9) and interpret the results.

4. Estimation and results

4.1 Population growth

The results for the population regression of equation (3-7) are presented in Table 4.1. The dependent

variable POPG is the difference in the population growth rate in a location from the average

population growth rate across all locations. Following Helpman (1998) and assuming validity of our

proxy variables we expect both of the independent variables to be positive and significant. Indeed

Table 4.1 shows significant coefficients of the expected positive sign. This suggests that both above

average access to goods and services (AGS) and above average inverse density (INVDENS) are

associated with an above average rate of population growth.

The theoretical interpretation of these results is based on how the dependent variables AGS and

INVDENS determine the relative utility between locations. In equilibrium, all locations offer the same

level of utility, since peripheral locations with worse access to goods and services ( or AGS) are

offset with a higher consumption of housing services ( or INVDENS). A reduction in the time cost of

transportation into a central business area increases the accessibility to goods and services for the

peripheral area, but has little effect on accessibility for the central area. Thus, the relative utility of

the peripheral region becomes higher and population grows faster there.

However in the NEG model the initial population movement is equivalent to a movement of firms

and in subsequent periods more goods and services can be sourced locally thus lowering expenditure

on transport and further raising relative utility. If the emergence of local business and services has a

positive effect on the utility of a location and this translates into higher level of population growth

then this would be captured in our empirical estimation as the AGS variable evolves over time.

Table 4.1: Empirical results - population regression

POPG

AGS 0.1969**

(6.35)

INVDENS 0 .0697**

(22.94)

CONST 0. 0263**

(16.52)

N 172271

R² 0.01 Notes: This table displays the parameter estimates for an OLS

regression of equation (3-7) rewritten to save space as

POPG = CONST + AGS + INVDENS +

t-statistics (in parenthesis) are heteroscedasticity robust.

** denotes significance at the 1% level.

16

This cumulative process eventually comes to an end in the theoretical model as peripheral areas

reach a certain density. Unfortunately this aspect of the model cannot be captured in the empirical

set up due to the fact there is no time dimension to the inverse density measure. As explained

earlier, the inverse density variable was averaged over time in an effort to remove endogeneity with

the dependent variable. In locations that grow significantly over the period, the proxy for inverse

density will be too small at the beginning and too large by the end. This sort of systematic error in an

explanatory variable is likely to have resulted in a loss of explanatory power and a bias in the

estimates.

The low R² in this model of 0.01 can be partially explained by the fact that the dependent variable is

both spatially and time-differenced. Nevertheless, it could also be a manifestation of the loss of

explanatory power as a result of the lack of time dimension for the inverse density measure.

Next we move on to the land values estimation that should not suffer from the same endogeneity

issues as the population regression, and can be estimated with a time dynamic inverse density

measure.

4.2 Land values

The results of the land value regressions are presented in Table 4.2. Column (1) gives the estimated

coefficients of specification (3-8) where we regress ’s difference from the mean land value in natural

logarithms on our explanatory variables. The positive coefficient of the AGS difference suggests that

a better access to a greater variety of goods at location compared with an average access is

associated with a relatively higher land value. That means better access to goods leads to an increase

in utility, which is equalised by higher land values. Conversely, inverse density is negatively correlated

with the land value difference. A location with a relatively low density (i.e. high inverse density) is

expected to have a lower land value. Based on the theory section, we would expect a positive

relationship between the land value difference and the difference in inverse density because density

is supposed to capture the dispersive force. However, the results of specification (3-8) suggest that

more people at location would lead to a higher land value and hence that the location is not dense

enough.

We report the results for specification (3-8) with controlled land values in column (2). Proximity to

green space, water bodies and industrial areas have been controlled for by using the residuals

obtained from equation (3-10) as dependent variable. The predicted results are similar to the ones in

the original specification. Both coefficients are smaller but maintain their original signs. The R² has

significantly decreased since basic amenities affecting the land values were already explained by the

auxiliary regression.

In column (3), distance to CBD has additionally been controlled for in the land value estimation. The

positive effect of the AGS difference has decreased while the difference in inverse density now has a

positive effect on land value differences at location . So lower density (higher inverse density) is

associated with higher land values which are expected to equalise the utility. Hence, when

controlling for centrality by including a land gradient in the first stage regression, increasing density

works as dispersive force.

17

The results of the growth specification (3-9) are illustrated in columns (4) and (5), controlling for

additional amenities with and without distance to CBD. While a relative high AGS at location is

associated with a negative land value growth, inverse density is positively related. Again, when

controlling for the CBD (5) both variables are positively correlated which is in line with our

expectations. Higher than average land value growth grown between and is associated with

above average AGS and inverse density measure in time period . This relationship illustrates the

adjustment process of the land values since we expect a more rapid increase in land values in

locations with a higher utility level. Moreover, the different estimation results suggest the inclusion

of a distance to CBD measure as a locational control that improves the estimation results in terms of

our expected coefficients. The low R² for the growth specifications can again be partially explained by

the spatial and time difference again.

These findings confirm our main hypothesis that with better access to a greater variety of goods,

triggered by lower transport costs, people appreciate a less dense location. If density at location is

relatively low (inverse density is high), the empirical results suggest higher land values at . Coupled

with the findings of the population regressions, people leave the dense historical CBD and move to

areas where they are able to (i) consume a greater amount of space and (ii) still have access to a high

variety of goods. Based on our historical sample for Berlin, infrastructure improvements that lower

the costs of transport raise relative utility in the periphery and foster an earlier relocalisation process

as also suggested by Anas (1992).

Table 4.2: Empirical results - land value estimations

(1) LV (2) LVresid (3) LVresid (4) LVresidG (5) LVresidG

AGS 9.0433** 5.7299** 1.1393** -0.0996** 0.4023**

(245.13) (149.15) (44.70) (-3.17) (20.93)

INVDENS -0.0716** -0.1358** 0.0285** 0.0193** 0.0142**

(-28.69) (-58.68) (16.94) (9.99) (10.52)

CONST -0.5721** -0.0554** 0.0064** 0.0417** -0.0057**

(-247.58) (-23.98) (4.03) (27.47) (-4.97)

Controls - yes yes yes yes

distance to CBD - - yes - yes

N 141825 141825 141825 129056 129056

R² 0.3767 0.2165 0.195 0.001 0.005 Notes: This table displays the parameter estimates for an OLS regression of equation (3-8) and (3-9) rewritten to save

space as LV = CONST + AGS + INVDENS + , whereas LV represents all the dependent variables depending on

the specification. t-statistics (in parenthesis) are heteroscedasticity robust. ** denotes significance at the 1% level.

The comparison of controlled specifications (3-8) and (3-9) suggests that the CBD still comprises

some centripetal forces which cannot be captured by our AGS indicator. The AGS indicator only

measures the shortest route between each population plot and each business/mixed use area. It

does not take into account the fact that many businesses may be visited in a single trip and thus puts

no extra weight on clusters of business locations. Furthermore, the quality of goods maybe better at

the CBD due to a higher productivity induced by agglomeration economies. Therefore, AGS is only an

accessibility indicator which lacks of describing potential agglomeration effects.

18

5. Conclusion

We have examined the problem of over-concentration in city centres using our case study of Berlin

1881-1914. The theoretical literature suggested that growing cities can become over-congested

because self-motivated individuals do not find it beneficial to locate anywhere other than the

business core. Therefore the problem is posed as one of co-ordination since the peripheral locations

do not offer a high enough level of utility until a significant number of individuals reside there. It was

suggested that transport infrastructure may play a key role in the determination of the utility of

locations within a city and could be used as a policy intervention in decongesting the city centre.

Our case of Berlin offers a unique historical point of view of a city characterised by both a rapidly

growing population and a huge development of its urban transit system. Making use of several

historical data sources, we have shown that the relative desirability of central and peripheral

locations is significantly determined by their relative accessibility to goods and services and the

relative congestion of space, as hypothesised in the NEG model constructed by Helpman (1998). In

addition, in the modelling of accessibility to goods and services we offered a unique interpretation of

the decay parameter as a measure of substitutability of differentiated goods.

Our results suggest that improving transport infrastructure may indeed provide a policy tool to aid

decongestion of city centres. It is important to note, though, that this may not be the correct policy

in all cases and that our results do not comment on the desirable level of agglomeration. Counter-

factual logic suggests that policymakers pursuing the opposite objective, of increasing concentration

in the CBD, should in fact avoid transport improvements that reduce the cost of travel to more

peripheral locations.

One important point is that the theory assumes that population movements are accompanied with a

proportional movement of firms. It has been suggested that if this is the case that our parameters

estimates would have captured that affect. However the phenomenon was neither modelled

explicitly in our empirical analysis nor discussed in detail in the theoretical literature. The process by

which the location of population drives the location decision of firms is a possible area for further

research.

19

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