=Paper=
{{Paper
|id=Vol-1627/paper6
|storemode=property
|title=Finding the Sweet Spot in the City: a Monopolistic Competition Approach
|pdfUrl=https://ceur-ws.org/Vol-1627/paper6.pdf
|volume=Vol-1627
|authors=Elizaveta Bespalova,Alim Moskalenko,Alexander Safin,Constantine Sorokin,Andrey Yagolkovsky
|dblpUrl=https://dblp.org/rec/conf/cla/BespalovaMSSY16
}}
==Finding the Sweet Spot in the City: a Monopolistic Competition Approach==
https://ceur-ws.org/Vol-1627/paper6.pdf
Finding the Sweet Spot in the City:
a Monopolistic Competition Approach
Elizaveta Bespalova, Alim Moskalenko, Alexander Safin,
Constantine Sorokin? , and Andrey Yagolkovsky
National Research University Higher School of Economics,
20 Myasnitskaya ulitsa,
Moscow 101000 Russia.
{csorokin}@hse.ru
http://www.hse.ru
Abstract. We propose a general equilibrium model to study the spatial
inequality of consumers and firms within a city. Our mechanics rely on
Dixit and Stiglitz monopolistic competition framework. The firms and
consumers are continuously distributed across a two-dimensional space,
there are iceberg-type costs both for goods shipping and workers com-
muting (hence firms have variable marginal costs based on their location).
Our main interest is in the equilibrium spatial distribution of wealth. We
construct a model that is both tractable and general enough to stand the
test of real city empirics. We provide some theoretical statements, but
mostly the results of numerical simulations with the real Moscow data.
Keywords: spatial distribution, linear city, circular city, monopolistic
competition
1 Introduction
We propose a general equilibrium model to study the spatial inequality of con-
sumers and firms within a city. Our mechanics rely on Dixit and Stiglitz mo-
nopolistic competition framework [1]. The firms and consumers are continuously
distributed across a two-dimensional space, there are iceberg-type costs (as in
Krugman’s models of trade[2]) both for goods shipping and workers commuting
(hence firms have variable marginal costs based on their location, we borrow
some of Melitz’s ideas here [6]). Our main interest is in the equilibrium spa-
tial distribution of wealth. We construct a model that is both tractable and
general enough to stand the test of real city empirics. We provide some theo-
retical statements, but mostly the results of numerical simulations with the real
?
The article was prepared within the framework of the Basic Research Program at
the National Research University Higher School of Economics (HSE) and supported
within the framework of a subsidy by the Russian Academic Excellence Project
’5-100’
74 Elizaveta Bespalova et al.
Moscow data. Our model is somewhat similar to [7], however, we rely on a dif-
ferent framework, adjusting the ideas from the international trade models to the
scale of a city.
Our theoretical modelling process consists of two parts. First, we developed
a one-dimensional model of the linear city on the [0, 1] interval. This model is an
adaptation of the Dixit-Stiglitz-Krugman model of trade to the case of spatial
distribution of firms and consumers within the city. Although it is far from
reflecting the real city structure, it is easier to solve and interpret the results.
Second, we demonstrate how the unidimensional model can be upgraded to the
empirics-friendly two-dimensional setup.
2 One-dimensional City
Consider a city with workers distributed uniformly on [0, 1] and firms distributed
uniformly on 12 − N2 ; 12 + N2 , where N is an endogenous parameter for the to-
tal number of the firms within the city. Each firm offers a single variety of a
composite differentiated good. We assume that the demand for that goods is not
only due to workers spending their wages, but also due to firm owners spending
their profits, so the consumers (both “firms” and workers) are distributed on
min 0, 21 − N2 ; max 1, 12 + N2 , which is hereinafter denoted by χ.
Consumer located at point x (or just consumer x) has an endowment of
e(x). Firm located at point y offers its own variety of composite differentiated
good at the price p(y). So here the product differentiation coincides with spatial
differentiation of firms — there is just one firm standing on the head of a pin.
Each consumers preferences are given by a CES-type utility function, thus they
exhibit the preference for variety. By solving the consumer’s problem we obtain
the demand q(x, y) — the amount of good that consumer x wants to buy from
firm y.
Our approach to firm’s production differs from the classical one in the struc-
ture of marginal costs. We assume that a firm has an equal probability of hiring
any worker in the city — firms can’t discriminate workers by their location, thus
there is a uniform equilibrium wage w. So in our model workers do not choose
place, where they want to work and wage, that they want to earn - place, where
they will work chooses randomly and wage is the same for all workers. Workers
commuting costs are paid by the firms, so that distant commuting results in
low productivity. Thus we have the following formula for labor requirements to
produce a unit of good for a firm located at point y:
Z1
c(y) = a (1 + θρ(x, y))dx,
0
where ρ(x, y) is a distance from point x to point y. Firms also have to pay a
fixed cost of f c(y)w.
The equilibrium in our model is characterized by the following conditions:
Finding the Sweet Spot in the City: a Monopolistic Competition Approach 75
1. Full employment: all the labor supplied is used in production. Let C(y) be
the total labor used by a firm located at y, so that we have:
1+N
Z2
C(y)dy = 1.
1−N
2
2. Free entry: firms located on the borders of the city have zero profits. Let
Q(y) be the overall amount of goods sold by firm y, therefore we have:
(p (N −1/2) − wc (N −1/2)) Q (N −1/2) − wc(y)f =
(p (N +1/2) − wc (N +1/2)) Q (N +1/2) − wc(y)f = 0.
3. Finally, the budget balance: the total expenditure at some point equals the
sum of worker’s wage and firm’s profits — of course, if they are located there.
Thus:
/ [0; 1] ∪ 21 − N2 ; 21 + N2 ,
0, y∈
y ∈ [0; 1] \ 12 − N2 ; 12 + N2 ,
w,
e(y) =
y ∈ 21 − N2 ; 21 + N2 \ [0; 1],
(p (y) − wc (y)) Q (y) − wc(y)f,
w + (p (y) − wc (y)) Q (y) − wc(y)f, y ∈ [0; 1] ∩ 21 − N2 ; 21 + N2 .
3 Results for One-dimensional City
We are able to prove the equilibrium existence result for the model, also we do
some natural comparative statics. However, for illustrative purposes we created
a MATLAB program to find the equilibrium given all the exogenous parameters
and illustrate all the comparative statics of interest. For example, for the follow-
ing values f = 0.2, a = 1, σ = 2, τ = 0.1, θ = 0.1, L = 0.1 the total number of
firms is 2.25 and the corresponding interval is [−0.625, 1.625].
For these values of parameters we can look at the changes in the profit of
firms π(y) and the optimal number of firms N according to some changes in
exogenous parameters. For all of the following graphs horizontal axis shows the
interval for firms and vertical axis shows the profits value. Blue lines here in after
depict small values of the changing parameter, yellow lines — highest values.
For instance, figure 1 illustrates that when the fixed cost for the firms f rises,
the optimal number of firms in the city falls, but the profit level of existing firms
π(y) in some times increases, but then falls with the rise of fixed costs. This is
happening because when the fixed costs rise for sufficiently low number of firms,
the effect from rising costs exceeds the one from “killing” competitors, so the
profit falls.
76 Elizaveta Bespalova et al.
Fig. 1. Firm profits as f changes from 0.1 to 0.2.
As we can see from figure 2, increase in the sensitivity to changes in distance
for the transportation of goods (τ ) and workers (θ) affect the profit of firms
differently. Although the number of firms N decreases only slightly with the rise
of both τ and θ, it causes a significant increase in profit, which is 1.5 times
greater for θ.
τ θ
Fig. 2. Firm profits as iceberg costs coefficient changes from 0.1 to 1.
Figure 3 shows us, that increase in elasticity of substitution (σ) leads to
decrease in number of firms and in their profit. It happens because in this case
preference for variety also decreases, so consumers prefer to buy goods from
neighbours and the farthest firms lose their profit.
Finding the Sweet Spot in the City: a Monopolistic Competition Approach 77
Fig. 3. Firm profits as σ changes from 1.5 to 3.5.
As we can see from figure 4, increase in number of workers gives us decrease
in profit for firms. It happens because bigger part of all money in the city goes
to wages for workers instead of going to profit for firms (total amount of money
in the city is constant and equals to 1)
Fig. 4. Firm profits as L changes from 0.5 to 1.5.
78 Elizaveta Bespalova et al.
4 Two-dimensional City
The generalisation of the model to the 2 dimensional case is straightforward —
one just needs to replace the unit interval with some compact set in R2 and
the Euclidean distance with some other metric that reflects the structure of
transportation within the city. The assumption that each point can host just
one firm cam be relaxed with some more general capacity constraint, but then
we will have also take into account that several varieties might be produced in a
single point, instead of just one — technically, it adds just one more dimension
to our model.
To adjust the model to the empirical estimation we assume that city is divided
into M districts, residents are distributed among districts and so do the firms;
each district has its own number of workers and a capacity constraint for firms.
Also, firms may be located on the radial highways spanning outside from the
borders of the city — we need this assumption to be able balance the number
of firms with free-entry condition. All the calculations are rather similar to the
continuous model, however, we use sums across districts to approximate the
integrals across space.
The main difference lies in the form of Melitz-inspired cutoff level condition.
We assume that firms, as they enter the market, first fill all the vacant offices
inside the city, from best to worst, and then the remaining ones locate along the
highways. So, if we have an exogenous value of the maximum number of firms
in each district Nj , then we can write the cutoff level condition for each of the
highways in linear form as in the one-dimensional model.
5 Finding real data for two-dimensional city
In our model we need to get some statistics about the city, such as distribution of
workers, distribution of firms, distances between objects in the city. We get real
data about Moscow, making detalization for districts. Distribution of workers we
get from official statistics about population in Moscow. Distribution of firms we
get by extrapolation of data from one non-official source, and distances between
districts we estimated, using specially designed algorithm.
The results of model will be distribution of profit among all firms in city,
value of wage and number of firms, which in model means length of highways.
To compare these results with real situation, we also get statistics about profit
for firms in all districts, using data about tax on profit from tax service.
6 Results for Two-dimensional City
Our next goal was to find out the values of exogenous parameters (elasticity of
substitution, for example) such that the results of the model closer to real data.
We were unable to run the classical OLS, instead, we varied our exogenous
parameters to make ratio between total profit of firms and total income of work-
ers equal to the real ratio in Moscow. It happens for a set of parameters, so,
Finding the Sweet Spot in the City: a Monopolistic Competition Approach 79
after that we tried to match the length of “occupied” highways and maximise
the Spearman coefficient that shows correlation between lists of districts, sorted
by profit for one firm, from model and from real data.
In the end, we get that in all sets f and σ are inversely dependent, τ and θ
are close to 1, Spearman coefficient is close to 0.62 and length of highways are
at most 25 km.
Also our results can be illustrated by figure 5, where districts are marked
with blue, if profit for conventional unit of firms there is bigger than income for
conventional unit of workers, else district is marked with yellow.
Fig. 5. Comparing profits of firms and income of workers
In figure 6 we illustrate length of
highways that we get from the model.
Fig. 6. Length of the highways
80 Elizaveta Bespalova et al.
And in figure 7 we compare distribution of profit for one firm between results
from model and real data.
Model Real data
Fig. 7. Distribution of profit for firms
Conclusion
To summarize, we see our main contribution in developing a model that is both
simple enough to be tractable and scalable enough to allow estimations using
real city data. The key feature is our ability to incorporate real city travel time
costs into a classical new economic geography mathematical framework. This
model can be used to address a variety of questions, from estimating economic
advantages of a particular location to evaluating the best directions of a long-
term city development. Though in this paper we mainly demonstrate the results
of model simulations, our efforts lead way to testing the model against a reality
of a modern city.
Finding the Sweet Spot in the City: a Monopolistic Competition Approach 81
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