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 Ak denote a city with a native job market of size k and Bn a city with an immigrant job market of size n. Then let ak(t) be the number of city Ak at time t and bn(t) be the number of city Bn at time t. The change rate of these two kinds of cities according to time t can be represented by the following equations:
(1)dak(t)dt=ak+1(t)∑i=1∞K1(k+1,i)ai(t) +ak-1(t)∑i=1∞K1(i,k-1)ai(t) -ak(t)∑i=1∞[K1(k,i)+K1(i,k)]ai(t) -ak(t)∑i=1∞I(k,i)bi(t)+ak-1(t)∑i=1∞I(k-1,i)bi(t) +ak+1(t)∑i=1∞J1(k+1,i)bi(t) -ak(t)∑i=1∞J1(k,i)bi(t),dbn(t)dt=bn+1(t)∑i=1∞K2(n+1,i)bi(t) +bn-1(t)∑i=1∞K2(i,n-1)bi(t) -bn(t)∑i=1∞[K2(n,i)+K2(i,n)]bi(t) +bn-1(t)∑i=1∞J2(n-1,i)ai(t) -bn(t)∑i=1∞J2(n,i)ai(t).
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 Ak, and K2(k,j) is the one for immigrant job markets. For convenience, we assume that the rate of a job movement from Aj to Ak 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 Bi to Ak, while J1(k,i) is the rate of job loss for Ak because of Bi, and J2(k,i) is the rate of job loss for Bk due to Ai. 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
(2)dak(t)dt =K1M1A(t)[(k+1)ak+1(t)+(k-1)ak-1(t)-2kak(t)] +IM1B(t)[(k-1)ak-1(t)-kak(t)] +JM1B(t)[(k+1)ak+1(t)-kak(t)],dbn(t)dt =K2M1B(t)[(n+1)bn+1(t)+(n-1)bn-1(t)-2nbn(t)] +JM1A(t)[(n-1)bn-1(t)-nbn(t)],
where M1A(t)=∑i=1∞iai(t) and M1B(t)=∑i=1∞ibi(t) are the total number of job positions of natives and immigrants at time t, respectively. M0A(t)=∑i=1∞ai(t) is the total number of cities with a native job market at time t. M0B(t)=∑i=1∞bi(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
(3)ak(t)=A(t)[a(t)]k-1, bn(t)=B(t)[b(t)]n-1,
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:
(4)da(t)dt=[K1M1A(t)(1-a(t))+IM1B(t)-JM1B(t)a(t)] ×(1-a(t)),dA(t)dt=-[2K1M1A(t)(1-a(t))+IM1B(t)+JM1B(t) -2JM1B(t)a(t)]A(t),db(t)dt=K2M1B(t)(1-b(t))2+JM1A(t)(1-b(t)),dB(t)dt=-B(t)[2K2M1B(t)(1-b(t))+JM1A(t)].
We note that M0A(t)=∑i=1∞ai(t)=A(t)/(1-a(t)) and M1A(t)=∑i=1∞iai(t)=A(t)/(1-a(t))2, then M0A(t)=M1A(t)(1-a(t)). Similarly M0B(t)=∑i=1∞bi(t)=B(t)/(1-b(t)), M1B(t)=∑i=1∞ibi(t) = B(t)/(1-b(t))2 and M0B(t)=M1B(t)(1-b(t)).
Using these moment expressions, system (4) can be rewritten into the following equations:
(5)11-a(t)da(t)dt=K1M0A(t)+IM1B(t)-JM1B(t)a(t),1A(t)dA(t)dt =-2K1M0A(t)-IM1B(t)+JM1B(t)(2a(t)-1),11-b(t)db(t)dt=K2M0B(t)+JM1A(t),1B(t)dB(t)dt=-2K2M0B(t)-JM1A(t).
The initial condition is
(6)a(0)=0, A(0)=A0,b(0)=0, B(0)=B0, at t=0.
From these equations, we can derive the following equations:
(7)dlnM0A(t)dt=11-a(t)da(t)dt+1A(t)dA(t)dt=-K1M0A(t)-JM1B(t)M0A(t)M1A(t),dlnM0B(t)dt=11-b(t)db(t)dt+1B(t)dB(t)dt=-K2M0B(t),dlnM1A(t)dt=21-a(t)da(t)dt+1A(t)dA(t)dt=(I-J)M1B(t),dlnM1B(t)dt=21-b(t)db(t)dt+1B(t)dB(t)dt=JM1A(t).
The relation between M1A(t) and M1B(t) can be derived directly under the initial condition (6); that is,
(8)J(M1A(t)-A0)=(I-J)(M1B(t)-B0).
With this relation, from (7) and (8), one can obtain
(9)dlnM1A(t)dt=J(M1A(t)-A0)+(I-J)B0,
that is,
(10)dM1A(t)dt=J(M1A(t))2+[(I-J)B0-JA0]M1A(t).
(I) In the case of (I-J)B0≠JA0, this Bernoulli equation (10) can be solved under the initial condition (6) to yield
(11)M1A(t)=A0[JA0-(I-J)B0]JA0-(I-J)B0exp[JA0-(I-J)B0]t.
The total number of immigrant job positions M1B(t) can be solved exactly in the same way from (7) and (11):
(12)M1B(t)=B0[(I-J)B0-JA0](I-J)B0-JA0exp[(I-J)B0-JA0]t.
Here, the solution is valid only in the time region t<t~, where t~=ln(JA0/(I-J)B0)/(JA0-(I-J)B0).
We can see that M1A(t) and M1B(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 t~, their kinetic behaviors can be analyzed. At a finite time t=t~, both M1A(t) and M1B(t) reach infinity, which implies that during a finite time there will be a tremendous increase of jobs.
In the time region t<t~, by substituting (11) and (12) into (7), one obtains
(13)1(M0A(t))2dM0A(t)dt=-K1-JB0A0exp[JA0-(I-J)B0]t.
Hence, we have
(14)M0A(t)=(A0[JA0-(I-J)B0]) ×(JA0-IB0+[JA0-(I-J)B0]K1A0t ×io+JB0exp[JA0-(I-J)B0]t)-1.
From (7), M0B(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 M1A(t) and M1B(t) go up. Meanwhile, M0A(t) goes down exponentially, and M0B(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
(15)a(t)=1-M0A(t)M1A(t)=1-(1-×oo+JB0exp[JA0-(I-J)B0]t)-1e899ee(JA0-(I-J)B0exp[JA0-(I-J)B0]t)1-oo×(JA0-IB0+[JA0-(I-J)B0]K1A0t1-×oo+JB0exp[JA0-(I-J)B0]t)-1).
From M0B(t)=B0(1+K2B0t)-1 and (12), one can obtain
(16)b(t)=1-M0B(t)M1B(t)=1-(I-J)B0-JA0exp[(I-J)B0-JA0]t[(I-J)B0-JA0](1+K2B0t).
Note that since ak(t)=A(t)[a(t)]k-1=M0A(t)(1-a(t))[a(t)]k-1 and bn(t)=B(t)[b(t)]n-1 = M0B(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<t~ as follows:
(17)ak(t)=(A0[JA0-(I-J)B0]×{JA0-(I-J)B0exp[JA0-(I-J)B0]t})×( +JB0exp[JA0-(I-J)B0]t}2{JA0-IB0+[JA0-(I-J)B0]K1A0t +JB0exp[JA0-(I-J)B0]t}2)-1×(1-(+JB0exp[JA0-(I-J)B0]t)-1)(JA0-(I-J)B0exp[JA0-(I-J)B0]t)×(1-)lo×(JA0-IB0+[JA0-(I-J)B0]K1A0t 1-×p+JB0exp[JA0-(I-J)B0]t)-1))k-1,bn(t)=(I-J)B02-JA0B0exp[(I-J)B0-JA0]t[(I-J)B0-JA0](1+K2B0t)2×(1-(I-J)B0-JA0exp[(I-J)B0-JA0]t[(I-J)B0-JA0](1+K2B0t))n-1.
When (I-J)B0≠JA0, the explicit forms of ak(t) and bn(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→∞, ak(t)→0 or when n→∞, bn(t)→0. These two extreme values of ak(t) and bn(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→t~, both ak(t)→0 and bn(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 M0A(t) and M0B(t).
(II) In the case of (I-J)B0=JA0, from (10), one obtains
(18)M1A(t)=A0(1-A0Jt)-1.
Substituting (18) into (7), one has
(19)M1B(t)=B0(1-A0Jt)-1.
Here, the solutions are valid only in the time region t<t-, where t-=(A0J)-1.
We can see that M1A(t) and M1B(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-, M1A(t) and M1B(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
(20)M0A(t)=A0[1+(A0K1+B0J)t]-1.
In addition, from (7), M0B(t)=B0[1+K2B0t]-1.
When (I-J)B0=JA0 holds, M1A(t) increases while M0A(t) decreases. The same happens to M1B(t) and M0B(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:
(21)ak(t)=(M0A(t))2M1A(t)(1-M0A(t)M1A(t))k-1=A0(1-A0Jt)[1+(A0K1+B0J)t]2×(1-1-A0Jt1+(A0K1+B0J)t)k-1,bn(t)=(M0B(t))2M1B(t)(1-M0B(t)M1B(t))n-1=B0(1-A0Jt)(1+K2B0t)2(1-1-A0Jt1+K2B0t)n-1.
These two explicit forms of ak(t) and bn(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, ak(t)→0 or when n→∞, ceteris paribus, bn(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 ak(t)→0 and bn(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.