We propose a general equilibrium model to study the spatial inequality of consumers and rms within a city. Our mechanics rely on Dixit and Stiglitz monopolistic competition framework. The rms and consumers are continuously distributed across a two-dimensional space, there are iceberg-type costs both for goods shipping and workers commuting (hence rms 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.
We propose a general equilibrium model to study the spatial inequality of
consumers and rms within a city. Our mechanics rely on Dixit and Stiglitz
monopolistic competition framework [
Moscow data. Our model is somewhat similar to [
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 rms and consumers within the city. Although it is far from re ecting 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
Consider a city with workers distributed uniformly on [0; 1] and rms distributed uniformly on 21 N2 ; 12 + N2 , where N is an endogenous parameter for the total number of the rms within the city. Each rm o ers a single variety of a composite di erentiated good. We assume that the demand for that goods is not only due to workers spending their wages, but also due to rm owners spending their pro ts, so the consumers (both \ rms" and workers) are distributed on min 0; 21 N2 ; max 1; 21 + N2 , which is hereinafter denoted by .
Consumer located at point x (or just consumer x) has an endowment of e(x). Firm located at point y o ers its own variety of composite di erentiated good at the price p(y). So here the product di erentiation coincides with spatial di erentiation of rms | there is just one rm 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 rm y.
Our approach to rm's production di ers from the classical one in the structure of marginal costs. We assume that a rm has an equal probability of hiring any worker in the city | rms 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 rms, 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 rm located at point y: where (x; y) is a distance from point x to point y. Firms also have to pay a xed cost of f c(y)w.
The equilibrium in our model is characterized by the following conditions: 1. Full employment: all the labor supplied is used in production. Let C(y) be the total labor used by a rm located at y, so that we have: 1+N Z2 1 2N
C(y)dy = 1: 2. Free entry: rms located on the borders of the city have zero pro ts. Let Q(y) be the overall amount of goods sold by rm 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 rm's pro ts | of course, if they are located there. Thus: e(y) = 80; > > ><w; >(p (y) wc (y)) Q (y) wc(y)f; > >:w + (p (y) wc (y)) Q (y) 1 y 2= [0; 1] [ 2 y 2 [0; 1] n 12 N2 ; 12 + N2 ; y 2 21 N2 ; 12 + N2 n [0; 1];
1 wc(y)f; y 2 [0; 1] \ 2
N2 ; 12 + N2 : N2 ; 12 + N2 ; 3
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 nd the equilibrium given all the exogenous parameters and illustrate all the comparative statics of interest. For example, for the following values f = 0:2; a = 1; = 2; = 0:1; = 0:1; L = 0:1 the total number of rms 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 pro t of rms (y) and the optimal number of rms N according to some changes in exogenous parameters. For all of the following graphs horizontal axis shows the interval for rms and vertical axis shows the pro ts value. Blue lines here in after depict small values of the changing parameter, yellow lines | highest values.
For instance, gure 1 illustrates that when the xed cost for the rms f rises, the optimal number of rms in the city falls, but the pro t level of existing rms (y) in some times increases, but then falls with the rise of xed costs. This is happening because when the xed costs rise for su ciently low number of rms, the e ect from rising costs exceeds the one from \killing" competitors, so the pro t falls.
As we can see from gure 2, increase in the sensitivity to changes in distance for the transportation of goods ( ) and workers ( ) a ect the pro t of rms di erently. Although the number of rms N decreases only slightly with the rise of both and , it causes a signi cant increase in pro t, which is 1.5 times greater for .
Figure 3 shows us, that increase in elasticity of substitution ( ) leads to decrease in number of rms and in their pro t. It happens because in this case preference for variety also decreases, so consumers prefer to buy goods from neighbours and the farthest rms lose their pro t.
changes from 1.5 to 3.5.
As we can see from gure 4, increase in number of workers gives us decrease in pro t for rms. It happens because bigger part of all money in the city goes to wages for workers instead of going to pro t for rms (total amount of money in the city is constant and equals to 1)
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 re ects the structure of transportation within the city. The assumption that each point can host just one rm 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 rms; each district has its own number of workers and a capacity constraint for rms. Also, rms 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 rms 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 di erence lies in the form of Melitz-inspired cuto level condition. We assume that rms, as they enter the market, rst ll all the vacant o ces 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 rms in each district Nj , then we can write the cuto level condition for each of the highways in linear form as in the one-dimensional model. 5
In our model we need to get some statistics about the city, such as distribution of workers, distribution of rms, distances between objects in the city. We get real data about Moscow, making detalization for districts. Distribution of workers we get from o cial statistics about population in Moscow. Distribution of rms we get by extrapolation of data from one non-o cial source, and distances between districts we estimated, using specially designed algorithm.
The results of model will be distribution of pro t among all rms in city, value of wage and number of rms, which in model means length of highways. To compare these results with real situation, we also get statistics about pro t for rms in all districts, using data about tax on pro t from tax service. 6
Our next goal was to nd 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 pro t of rms and total income of workers equal to the real ratio in Moscow. It happens for a set of parameters, so, after that we tried to match the length of \occupied" highways and maximise the Spearman coe cient that shows correlation between lists of districts, sorted by pro t for one rm, from model and from real data.
In the end, we get that in all sets f and are inversely dependent, and are close to 1, Spearman coe cient is close to 0.62 and length of highways are at most 25 km.
Also our results can be illustrated by gure 5, where districts are marked with blue, if pro t for conventional unit of rms there is bigger than income for conventional unit of workers, else district is marked with yellow.
In gure 6 we illustrate length of highways that we get from the model.
And in gure 7 we compare distribution of pro t for one rm between results from model and real data.
Model
Real data
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 longterm city development. Though in this paper we mainly demonstrate the results of model simulations, our e orts lead way to testing the model against a reality of a modern city.