Migration-Driven Aggregation Behaviors of Job Markets in a Multi-Group Environment

Abstract

This paper introduces a new model describing the aggregate growth of job markets. We divide the job market in each city into two groups: native job market of size 𝑘 and an immigrant job market of size 𝑛. A reversible migration of jobs exists in both groups. In addition, the interaction between these two groups creates both native and immigrant jobs. A loss of native jobs also takes place due to the interaction. Through studying initial conditions, job-creation rate, and job-loss rate we discover some meaningful results. The size change of native job market is closely related to that of the migration rate, native job-creation rate, and native job-loss rate. We assume that these rates are proportional to the sizes of two groups and find out that for certain initial conditions, immigrants influence native job markets positively. They create more jobs for both job markets. In addition, we can make conclusions about the future trend of the flow of jobs. People will move to places like big cities where there is a higher concentration of job opportunities.

1. Introduction

Much research has been done on the aggregation process in the past few decades. This enthusiasm starts from natural scientists, according to [1–3] and references therein. Biologists find it possible to study the behavior of species via aggregation process [2], and physicists use it to look at dynamics of different systems [1, 3]. With the development of knowledge about networks, aggregation process has been applied to solve many networking problems. These problems vary from population migration [4] to social network about individual interactions in [5, 6]. Some people have introduced this method to social sciences as well, such as [7, 8].

Another field where aggregation process is vastly applied is population dynamics [9–11]. Population groups are viewed as aggregates, and the interaction among them can be assimilated to the ones among aggregates with different sizes. Immigration phenomenon is studied through aggregation process as well, such as in [12, 13]. Here immigration leads to two major effects: catalyzed birth and death rates on the original group. The relative dominance of one effect over the other is analyzed through the growth of the aggregates.

Some assume an irreversible reaction scheme when dealing with aggregates: 𝐴_𝑘+𝐴−−−−−→_𝑙^{𝐾(𝑘,𝑙)}𝐴_𝑘−1+𝐴_𝑙+1(𝑘≤𝑙) in [1]. Here, 𝐴_𝑘 is an aggregate 𝐴 with size 𝑘, and 𝐾(𝑘;𝑙) is the rate of migration from 𝐴_𝑘 to 𝐴_𝑙. This scheme shows that for each individual monomer, it prefers larger aggregates and always moves from the smaller ones to the bigger aggregates. However, a more general reversible reaction process exists as well where each monomer moves from bigger aggregates to smaller ones in [14].

Job is an important topic in labor economics. There are many different approaches to this issue; for example, [15] addressed the influence of social networks on job market transitions and [16] looked at the process of job searching. Recently, scholars have combined the issue of job with population migration. In [17], the author revisited the area approach by analyzing the effects of immigrants on the labor demand for natives in the US. Reference [18] provided a dynamic structural model of migration decisions that is aggregated to describe the behavior of interregional migration. The author proposed a new job model with migration-driven aggregation process, and investigated the evolution of job markets of different sizes in multi-link cities in [19]. Motivated by the above analysis, in our paper, we investigate the evolution of the job markets of different sizes in cities. To the best of our knowledge, there are a few results in the literature on this topic.

In this paper, we divide the job market in cities into two groups: 𝐴_𝑘 means a city that has a job market with 𝑘 positions occupied by natives; 𝐵_𝑛 is a city with a job market of size 𝑛 for immigrants. There exists a migration of jobs among job markets of the same kind. This migration is reversible as jobs move from small cities to big ones and vice versa. In addition, various 𝐴_𝑘 and 𝐵_𝑛 interact with each other as well. The presence of immigrants in the job markets has both positive and negative impacts on the original one.

2. Model and Analysis

First, we make an assumption about this model. The job markets are spatially homogeneous. That means their only difference lies in their sizes.

Let 𝐴_𝑘 denote a city with a native job market of size 𝑘 and 𝐵_𝑛 a city with an immigrant job market of size 𝑛. Then let 𝑎_𝑘(𝑡) be the number of city 𝐴_𝑘 at time 𝑡 and 𝑏_𝑛(𝑡) be the number of city 𝐵_𝑛 at time 𝑡. The change rate of these two kinds of cities according to time 𝑡 can be represented by the following equations: 𝑑𝑎_𝑘(𝑡)𝑑𝑡=𝑎_𝑘+1(𝑡)^∞_𝑖=1𝐾₁(𝑘+1,𝑖)𝑎_𝑖(𝑡)+𝑎_𝑘−1(𝑡)^∞_𝑖=1𝐾₁(𝑖,𝑘−1)𝑎_𝑖(𝑡)−𝑎_𝑘(𝑡)^∞_𝑖=1𝐾₁(𝑘,𝑖)+𝐾₁(𝑖,𝑘)𝑎_𝑖(𝑡)−𝑎_𝑘(𝑡)^∞_𝑖=1𝐼(𝑘,𝑖)𝑏_𝑖(𝑡)+𝑎_𝑘−1(𝑡)^∞_𝑖=1𝐼(𝑘−1,𝑖)𝑏_𝑖(𝑡)+𝑎_𝑘+1(𝑡)^∞_𝑖=1𝐽₁(𝑘+1,𝑖)𝑏_𝑖(𝑡)−𝑎_𝑘(𝑡)^∞_𝑖=1𝐽₁(𝑘,𝑖)𝑏_𝑖(𝑡),𝑑𝑏_𝑛(𝑡)𝑑𝑡=𝑏_𝑛+1(𝑡)^∞_𝑖=1𝐾₂(𝑛+1,𝑖)𝑏_𝑖(𝑡)+𝑏_𝑛−1(𝑡)^∞_𝑖=1𝐾₂(𝑖,𝑛−1)𝑏_𝑖(𝑡)−𝑏_𝑛(𝑡)^∞_𝑖=1𝐾₂(𝑛,𝑖)+𝐾₂(𝑖,𝑛)𝑏_𝑖(𝑡)+𝑏_𝑛−1(𝑡)^∞_𝑖=1𝐽₂(𝑛−1,𝑖)𝑎_𝑖(𝑡)−𝑏_𝑛(𝑡)^∞_𝑖=1𝐽₂(𝑛,𝑖)𝑎_𝑖(𝑡). (1)

The first three terms in both equations are the migration within the same group. Here, 𝐾₁(𝑘,𝑗) is the exchange rate of jobs among different 𝐴_𝑘, and 𝐾₂(𝑘,𝑗) is the one for immigrant job markets. For convenience, we assume that the rate of a job movement from 𝐴_𝑗 to 𝐴_𝑘 is proportional to the size 𝑗 and 𝑘. We denote 𝐾₁(𝑘,𝑗)=𝐾₁𝑘𝑗, 𝐾₂(𝑘,𝑗)=𝐾₂𝑘𝑗. This means the bigger these two job markets are, the more frequent job movement takes place. This makes sense in real life as the migration of jobs happens much more frequently in job concentrated areas.

In addition to migrations within the same group, the existence of immigrant job market has two effects on the native one. First, the transfer of immigrant jobs might create new job positions for the natives. However, another effect is the opposite. The influx of immigrants might take over the jobs of natives. Then natives either have to look for a job in another city or become unemployed. The fourth and fifth terms in the first equation describe the jobs created through the interaction of two groups. The last two terms in the first equation describe the job loss due to the interaction. 𝐼(𝑘,𝑖) represents the rate of jobs created due to 𝐵_𝑖 to 𝐴_𝑘, while 𝐽₁(𝑘,𝑖) is the rate of job loss for 𝐴_𝑘 because of 𝐵_𝑖, and 𝐽₂(𝑘,𝑖) is the rate of job loss for 𝐵_𝑘 due to 𝐴_𝑖. Here, we assume that every job lost by an immigrant is gained by a native, so 𝐽₁=𝐽₂. The last two terms in the second equation show the gain of jobs for the immigrants via the interaction with the natives. We further assume that 𝐼(𝑘,𝑖)=𝐼𝑘𝑖 and 𝐽₁(𝑘,𝑖)=𝐽₂(𝑘,𝑖)=𝐽𝑘𝑖, respectively.

The rate equations for our system (1) are reduced to 𝑑𝑎_𝑘(𝑡)𝑑𝑡=𝐾₁𝑀^𝐴₁(𝑡)(𝑘+1)𝑎_𝑘+1(𝑡)+(𝑘−1)𝑎_𝑘−1(𝑡)−2𝑘𝑎_𝑘(𝑡)+𝐼𝑀^𝐵₁(𝑡)(𝑘−1)𝑎_𝑘−1(𝑡)−𝑘𝑎_𝑘(𝑡)+𝐽𝑀^𝐵₁(𝑡)(𝑘+1)𝑎_𝑘+1(𝑡)−𝑘𝑎_𝑘(𝑡),𝑑𝑏_𝑛(𝑡)𝑑𝑡=𝐾₂𝑀^𝐵₁(𝑡)(𝑛+1)𝑏_𝑛+1(𝑡)+(𝑛−1)𝑏_𝑛−1(𝑡)−2𝑛𝑏_𝑛(𝑡)+𝐽𝑀^𝐴₁(𝑡)(𝑛−1)𝑏_𝑛−1(𝑡)−𝑛𝑏_𝑛(𝑡), (2) where 𝑀^𝐴₁(𝑡)=∑^∞_𝑖=1𝑖𝑎_𝑖(𝑡) and 𝑀^𝐵₁(𝑡)=∑^∞_𝑖=1𝑖𝑏_𝑖(𝑡) are the total number of job positions of natives and immigrants at time 𝑡, respectively. 𝑀^𝐴₀(𝑡)=∑^∞_𝑖=1𝑎_𝑖(𝑡) is the total number of cities with a native job market at time 𝑡. 𝑀^𝐵₀(𝑡)=∑^∞_𝑖=1𝑏_𝑖(𝑡) is the total number of cities with an immigrant job market at time 𝑡.

In this paper, we find that our current rate equations (2) can be solved by the Ansatz [4, 7]. Consider 𝑎_𝑘(𝑡)=𝐴(𝑡)[𝑎(𝑡)]^𝑘−1,𝑏_𝑛(𝑡)=𝐵(𝑡)[𝑏(𝑡)]^𝑛−1, (3) where 𝐴(𝑡), 𝐵(𝑡), 𝑎(𝑡), and 𝑏(𝑡) are continuous functions, and |𝑎(𝑡)|<1, |𝑏(𝑡)|<1.

By substituting Ansatz (3) into the rate equation (2), it can be transformed into the differential equations as follows: 𝑑𝑎(𝑡)=𝐾𝑑𝑡¹𝑀^𝐴₁(𝑡)(1−𝑎(𝑡))+𝐼𝑀^𝐵₁(𝑡)−𝐽𝑀^𝐵₁(𝑡)𝑎(𝑡×((𝑡,(𝑡))1−𝑎))𝑑𝐴𝑑𝑡=−2𝐾₁𝑀^𝐴₁(𝑡)(1−𝑎(𝑡))+𝐼𝑀^𝐵₁(𝑡)+𝐽𝑀^𝐵₁(𝑡)−2𝐽𝑀^𝐵₁(𝑡)𝑎(𝑡𝐴(𝑡),(𝑡))𝑑𝑏𝑑𝑡=𝐾₂𝑀^𝐵₁(𝑡)(1−𝑏(𝑡))²+𝐽𝑀^𝐴₁(𝑡(𝑡,(𝑡))(1−𝑏))𝑑𝐵(𝑡𝑑𝑡=−𝐵)2𝐾₂𝑀^𝐵₁(𝑡)(1−𝑏(𝑡))+𝐽𝑀^𝐴₁(𝑡). (4)

We note that 𝑀^𝐴₀(𝑡)=∑^∞_𝑖=1𝑎_𝑖(𝑡)=𝐴(𝑡)/(1−𝑎(𝑡)) and 𝑀^𝐴₁(𝑡)=∑^∞_𝑖=1𝑖𝑎_𝑖(𝑡)=𝐴(𝑡)/(1−𝑎(𝑡))², then 𝑀^𝐴₀(𝑡)=𝑀^𝐴₁(𝑡)(1−𝑎(𝑡)). Similarly 𝑀^𝐵₀(𝑡)=∑^∞_𝑖=1𝑏_𝑖(𝑡)=𝐵(𝑡)/(1−𝑏(𝑡)), 𝑀^𝐵₁(𝑡)=∑^∞_𝑖=1𝑖𝑏_𝑖(𝑡) = 𝐵(𝑡)/(1−𝑏(𝑡))² and 𝑀^𝐵₀(𝑡)=𝑀^𝐵₁(𝑡)(1−𝑏(𝑡)).

Using these moment expressions, system (4) can be rewritten into the following equations: 1(𝑡)(𝑡)1−𝑎𝑑𝑎𝑑𝑡=𝐾₁𝑀^𝐴₀(𝑡)+𝐼𝑀^𝐵₁(𝑡)−𝐽𝑀^𝐵₁(𝑡)𝑎(𝑡),1𝐴(𝑡)𝑑𝐴(𝑡)𝑑𝑡=−2𝐾₁𝑀^𝐴₀(𝑡)−𝐼𝑀^𝐵₁(𝑡)+𝐽𝑀^𝐵₁(𝑡(𝑡)),1)(2𝑎−1(𝑡)(𝑡)1−𝑏𝑑𝑏𝑑𝑡=𝐾₂𝑀^𝐵₀(𝑡)+𝐽𝑀^𝐴₁(𝑡),1𝐵(𝑡)𝑑𝐵(𝑡)𝑑𝑡=−2𝐾₂𝑀^𝐵₀(𝑡)−𝐽𝑀^𝐴₁(𝑡). (5)

The initial condition is 𝑎(0)=0,𝐴(0)=𝐴₀,𝑏(0)=0,𝐵(0)=𝐵₀,at𝑡=0. (6)

From these equations, we can derive the following equations: 𝑑ln𝑀^𝐴₀(𝑡)=1𝑑𝑡(𝑡)(𝑡)1−𝑎𝑑𝑎+1𝑑𝑡𝐴(𝑡)𝑑𝐴(𝑡)𝑑𝑡=−𝐾₁𝑀^𝐴₀(𝑡)−𝐽𝑀^𝐵₁(𝑡)𝑀^𝐴₀(𝑡)𝑀^𝐴₁(𝑡),𝑑ln𝑀^𝐵₀(𝑡)=1𝑑𝑡(𝑡)(𝑡)1−𝑏𝑑𝑏+1𝑑𝑡𝐵(𝑡)𝑑𝐵(𝑡)𝑑𝑡=−𝐾₂𝑀^𝐵₀(𝑡),𝑑ln𝑀^𝐴₁(𝑡)=2𝑑𝑡(𝑡)(𝑡)1−𝑎𝑑𝑎+1𝑑𝑡𝐴(𝑡)𝑑𝐴(𝑡)=()𝑀𝑑𝑡𝐼−𝐽^𝐵₁(𝑡),𝑑ln𝑀^𝐵₁(𝑡)=2𝑑𝑡(𝑡)(𝑡)1−𝑏𝑑𝑏+1𝑑𝑡𝐵(𝑡)𝑑𝐵(𝑡)𝑑𝑡=𝐽𝑀^𝐴₁(𝑡). (7)

The relation between 𝑀^𝐴₁(𝑡) and 𝑀^𝐵₁(𝑡) can be derived directly under the initial condition (6); that is, 𝐽𝑀^𝐴₁(𝑡)−𝐴₀=(𝐼−𝐽)𝑀^𝐵₁(𝑡)−𝐵₀. (8)

With this relation, from (7) and (8), one can obtain 𝑑ln𝑀^𝐴₁(𝑡)𝑀𝑑𝑡=𝐽^𝐴₁(𝑡)−𝐴₀+(𝐼−𝐽)𝐵₀, (9) that is, 𝑑𝑀^𝐴₁(𝑡)𝑀𝑑𝑡=𝐽^𝐴₁(𝑡)²+(𝐼−𝐽)𝐵₀−𝐽𝐴₀𝑀^𝐴₁(𝑡). (10)

(I) In the case of (𝐼−𝐽)𝐵₀≠𝐽𝐴₀, this Bernoulli equation (10) can be solved under the initial condition (6) to yield 𝑀^𝐴₁(𝑡)=𝐴₀𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝐽𝐴₀−(𝐼−𝐽)𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡. (11)

The total number of immigrant job positions 𝑀^𝐵₁(𝑡) can be solved exactly in the same way from (7) and (11): 𝑀^𝐵₁(𝑡)=𝐵₀(𝐼−𝐽)𝐵₀−𝐽𝐴₀(𝐼−𝐽)𝐵₀−𝐽𝐴₀exp(𝐼−𝐽)𝐵₀−𝐽𝐴₀𝑡. (12)

Here, the solution is valid only in the time region 𝑡<𝑡̃, where 𝑡̃=ln(𝐽𝐴₀/(𝐼−𝐽)𝐵₀)/(𝐽𝐴₀−(𝐼−𝐽)𝐵₀).

We can see that 𝑀^𝐴₁(𝑡) and 𝑀^𝐵₁(𝑡) grow with time in the above cases, and they grow much faster in time because the rates of job migrating are both proportional to the size of the job market itself. When 𝑡 reaches 𝑡̃, their kinetic behaviors can be analyzed. At a finite time 𝑡=𝑡̃, both 𝑀^𝐴₁(𝑡) and 𝑀^𝐵₁(𝑡) reach infinity, which implies that during a finite time there will be a tremendous increase of jobs.

In the time region 𝑡<𝑡̃, by substituting (11) and (12) into (7), one obtains 1𝑀^𝐴₀(𝑡)²𝑑𝑀^𝐴₀(𝑡)𝑑𝑡=−𝐾₁−𝐽𝐵₀𝐴₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡. (13)

Hence, we have 𝑀^𝐴₀(𝑡)=𝐴₀𝐽𝐴₀−(𝐼−𝐽)𝐵₀×𝐽𝐴₀−𝐼𝐵₀+𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝐾₁𝐴₀𝑡+𝐽𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡⁻¹. (14)

From (7), 𝑀^𝐵₀(𝑡)=𝐵₀(1+𝐾₂𝐵₀𝑡)⁻¹.

Under the condition that (𝐼−𝐽)𝐵₀≠𝐽𝐴₀, we can see that the total number of national jobs for natives increases exponentially and the one for immigrants increases exponentially as well due to the fact that both 𝑀^𝐴₁(𝑡) and 𝑀^𝐵₁(𝑡) go up. Meanwhile, 𝑀^𝐴₀(𝑡) goes down exponentially, and 𝑀^𝐵₀(𝑡) decreases over time.

These mathematical results show that when the rates 𝐼, 𝐽, and the initial conditions satisfy the given inequality, the number of jobs for natives grows exponentially over time and the jobs for immigrants increase exponentially as well. Both the native and immigrant job markets enlarge in sizes; this means that more people are getting employed. In addition, since the number of cities for both groups goes down as time goes, there is a phenomenon that people will move to big cities to look for job opportunities. This leads to a concentration of jobs in big cities and gradually leaves many rural places empty since 𝑎(𝑡)𝑀=1−^𝐴0(𝑡)𝑀^𝐴₁(𝑡)=1−𝐽𝐴₀−(𝐼−𝐽)𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡×𝐽𝐴₀−𝐼𝐵₀+𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝐾₁𝐴₀𝑡+𝐽𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡⁻¹. (15)

From 𝑀^𝐵₀(𝑡)=𝐵₀(1+𝐾₂𝐵₀𝑡)⁻¹ and (12), one can obtain 𝑏(𝑡)𝑀=1−^𝐵0(𝑡)𝑀^𝐵₁(𝑡)()𝐵=1−𝐼−𝐽₀−𝐽𝐴₀exp(𝐼−𝐽)𝐵₀−𝐽𝐴₀𝑡(𝐼−𝐽)𝐵₀−𝐽𝐴₀1+𝐾₂𝐵₀𝑡. (16)

Note that since 𝑎_𝑘(𝑡)=𝐴(𝑡)[𝑎(𝑡)]^𝑘−1=𝑀^𝐴₀(𝑡)(1−𝑎(𝑡))[𝑎(𝑡)]^𝑘−1 and 𝑏_𝑛(𝑡)=𝐵(𝑡)[𝑏(𝑡)]^𝑛−1 = 𝑀^𝐵₀(𝑡)(1−𝑏(𝑡))[𝑏(𝑡)]^𝑛−1, we derive the kinetic behaviors of the number of jobs in two groups with the time constraint 𝑡<𝑡̃ as follows: 𝑎_𝑘(𝑡)=𝐴₀𝐽𝐴₀−(𝐼−𝐽)𝐵₀×𝐽𝐴₀−(𝐼−𝐽)𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡×𝐽𝐴₀−𝐼𝐵₀+𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝐾₁𝐴₀𝑡+𝐽𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡²⁻¹×1−𝐽𝐴₀−(𝐼−𝐽)𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡×𝐽𝐴₀−𝐼𝐵₀+𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝐾₁𝐴₀𝑡+𝐽𝐵₀exp𝐽𝐴₀−(𝐼−𝐽)𝐵₀𝑡^−1𝑘−1,𝑏_𝑛(𝑡)=(𝐼−𝐽)𝐵²₀−𝐽𝐴₀𝐵₀exp(𝐼−𝐽)𝐵₀−𝐽𝐴₀𝑡(𝐼−𝐽)𝐵₀−𝐽𝐴₀1+𝐾₂𝐵₀𝑡²×()𝐵1−𝐼−𝐽₀−𝐽𝐴₀exp(𝐼−𝐽)𝐵₀−𝐽𝐴₀𝑡(𝐼−𝐽)𝐵₀−𝐽𝐴₀1+𝐾₂𝐵₀𝑡^𝑛−1. (17)

When (𝐼−𝐽)𝐵₀≠𝐽𝐴₀, the explicit forms of 𝑎_𝑘(𝑡) and 𝑏_𝑛(𝑡) at time 𝑡 are as above. These solutions provide us with a clear and direct view at the number of cities with various sizes of job markets. Meanwhile, these two forms have further implications. Holding 𝑡 constant, when 𝑘→∞, 𝑎_𝑘(𝑡)→0 or when 𝑛→∞, 𝑏_𝑛(𝑡)→0. These two extreme values of 𝑎_𝑘(𝑡) and 𝑏_𝑛(𝑡) obviously are not within the obtainable range in real life, but they imply that at certain times, a city with a job market with infinite jobs does not exist. That is, every city has a job market with a size of a finite number. Another case is holding other factors constant; when 𝑡→𝑡̃, both 𝑎_𝑘(𝑡)→0 and 𝑏_𝑛(𝑡)→0. These two values describe the situation that during finite time periods, the number of cities with certain job markets goes to zero. The further explanation of this is that all people leave their local cities and move to big cities with more job opportunities. This conclusion is consistent with our analysis above about 𝑀^𝐴₀(𝑡) and 𝑀^𝐵₀(𝑡).

(II) In the case of (𝐼−𝐽)𝐵₀=𝐽𝐴₀, from (10), one obtains 𝑀^𝐴₁(𝑡)=𝐴₀1−𝐴₀𝐽𝑡⁻¹. (18) Substituting (18) into (7), one has 𝑀^𝐵₁(𝑡)=𝐵₀1−𝐴₀𝐽𝑡⁻¹. (19) Here, the solutions are valid only in the time region 𝑡<𝑡, where 𝑡=(𝐴₀𝐽)⁻¹.

We can see that 𝑀^𝐴₁(𝑡) and 𝑀^𝐵₁(𝑡) grow with time as in the previous cases, and they grow much faster because the job migrating rates are proportional to the size of the job markets. Their kinetic behaviors can be analyzed in the time region 𝑡<𝑡. When 𝑡 increases and reaches 𝑡, 𝑀^𝐴₁(𝑡) and 𝑀^𝐵₁(𝑡) approach infinite values. This implies a huge increase of jobs for both groups during this time period.

In the time region 𝑡<𝑡, by substituting (18) and (19) into (7), one obtains 𝑀^𝐴₀(𝑡)=𝐴₀1+𝐴₀𝐾₁+𝐵₀𝐽𝑡⁻¹. (20)

In addition, from (7), 𝑀^𝐵₀(𝑡)=𝐵₀[1+𝐾₂𝐵₀𝑡]⁻¹.

When (𝐼−𝐽)𝐵₀=𝐽𝐴₀ holds, 𝑀^𝐴₁(𝑡) increases while 𝑀^𝐴₀(𝑡) decreases. The same happens to 𝑀^𝐵₁(𝑡) and 𝑀^𝐵₀(𝑡); the first goes up while the second one goes down.

These changes show a migration of population into big cities where people can find more jobs due to the availability of opportunities, and the presence of immigrants in the country will create a win-win situation. More jobs will be offered to both natives and immigrants. This is the ideal situation where everybody benefits from this immigration of foreign population.

We obtain the kinetic behaviors of the number of job markets for natives and immigrants with the time constraint 𝑡<𝑡 as follows: 𝑎_𝑘(𝑡)=𝑀^𝐴₀(𝑡)²𝑀^𝐴₁(𝑡)𝑀1−^𝐴0(𝑡)𝑀^𝐴₁(𝑡)^𝑘−1=𝐴₀1−𝐴₀𝐽𝑡1+𝐴₀𝐾₁+𝐵₀𝐽𝑡²×1−1−𝐴₀𝐽𝑡1+𝐴₀𝐾₁+𝐵₀𝐽𝑡^𝑘−1,𝑏_𝑛(𝑡)=𝑀^𝐵₀(𝑡)²𝑀^𝐵₁(𝑡)𝑀1−^𝐵0(𝑡)𝑀^𝐵₁(𝑡)^𝑛−1=𝐵₀1−𝐴₀𝐽𝑡1+𝐾₂𝐵₀𝑡²1−1−𝐴₀𝐽𝑡1+𝐾₂𝐵₀𝑡^𝑛−1. (21)

These two explicit forms of 𝑎_𝑘(𝑡) and 𝑏_𝑛(𝑡) are different from the ones under the condition that (𝐼−𝐽)𝐵₀=𝐽𝐴₀. However, both of them show the same characteristics: when 𝑘→∞, ceteris paribus, 𝑎_𝑘(𝑡)→0 or when 𝑛→∞, ceteris paribus, 𝑏_𝑛(𝑡)→0. At a certain time point, a city can only have a job market with finite jobs. In addition, holding everything else equal, when 𝑡→𝑡, both 𝑎_𝑘(𝑡)→0 and 𝑏_𝑛(𝑡)→0. That is, within the finite time period, people will move to more job concentrated areas leaving other areas with finite jobs gradually empty.

3. Conclusions

We have introduced an aggregate growth model with job-creation rate, job-loss rate, and reversible migration rate via interaction into the job markets. By studying the kinetic behavior of native and immigrant job markets, we obtain some results about job migration during a certain finite time period.

Under the assumption that rate of job loss for native and rate of job gain for immigrants equal each other: 𝐽₁=𝐽₂, we can divide the solution of the original rate equations into two cases. First, if the initial conditions and positive and negative migration rates satisfy (𝐼−𝐽)𝐵₀≠𝐽𝐴₀, when time 𝑡 approaches 𝑡̃=ln(𝐽𝐴₀/(𝐼−𝐽)𝐵₀)/(𝐽𝐴₀−(𝐼−𝐽)𝐵₀), the total number of jobs for both native and immigrants increases exponentially. At the same time, the labor force becomes more concentrated for both groups as people tend to move to places with more job opportunities.

The second case takes place when the condition (𝐼−𝐽)𝐵₀=𝐽𝐴₀ is met. As time goes to 𝑡=(𝐴₀𝐽)⁻¹, the amount of jobs for both immigrants and natives grows exponentially as well. Moreover, the population of natives and immigrants starts to concentrate in limited cities. This leads to a map with people highly concentrated in cities and leaving other areas empty.

Through the analysis of the previous two cases, it is not hard to see that the results are consistent in both cases as immigrants bring a positive impact on native job markets. With the influx of immigrants, the positive impact dominates the migration rate of jobs. Our results are consistent with University of California, Berkeley's Professor David Card's finding that immigrants benefit the entire society by creating more jobs for both groups. This conclusion has meaningful implications for governmental policies. The analysis of this model helps explain the hotly debated immigration issue in economics and politics. Instead of putting stricter control on immigration or lowering the rate of immigration, some countries should think about the policies that encourage the inflow of immigrants since these immigrants actually have a positive impact on domestic job market.