Migration Models

Modeling of any process may involve two stages namely (i) theoretical statement about the process and (ii) translation of a statement into mathematical model. The theories of migration attempt to explain the reasons for migration. At this stage, an attempt is made to list out in fact the factors responsible for a person’s movement. In the second stage, researchers try to parameterize the process so that it can have wider applicability and help in planning. Following are the theories which have emerged from various studies on migration.

1. Ravenstein’s Law of Migration.
2. Zipf’s Gravity Model.
3. Lee’s Theory of Migration.
4. Todaro’s Model.
5. Stouffer’s Model.
6. L-F-R Model of Development theory.
7. Sjaatad’ Human Investment theory.
8. Cost-Benefit Model.
9. Mover-Stayer Model of Migration.

In addition to the above theories, which explain the flow from one area to another one, attempts have been made to develop mathematical model for the age distribution of migrants or special type of migration; for example models showing the relation of marriage with distance. It has been observed that age profile of the migrants exhibit a regular pattern. Rogers et.al. 1987 therefore adopted the notion of model migration schedules in mathematical form. This has also been illustrated by Roger and Castro 1981. In this case age pattern may be given as: 

It seems there are four curves representing different age segments of migration. They are: 

• Pre-labor force ages (negative exponential with ∞1 rate of descent).
• Labor force (double exponential, with mean age µ2, rate of ascent α2 and descent ƛa).
•  Post labor force (double exponential with mean age µ3 rate of ascent α3 and descent, ƛ3.
• A constant curve C.

For detail of this model and its utility one can see the given reference.

A model for marriage and distance was proposed by Morill and Pitts (1967) on the line of Zipf’s Gravity model. This model is given by 

M=a/D

Where M is migrants at distance D and a is a constant. It was observed that like Gravity model, above model tends to overestimate close in frequencies (Short distance). So they proposed the following modified model.

Recently Sharma (1984) and Yadav (1989) have proposed a number of modifications over the above model. These models are not given here because the quoted research work  is not widely available to the researchers.

We briefly discuss the theories of migration which are widely used in migration analysis:

A. Raventein’s Law of Migration: Ravenstein made the following propositions.

•  Migrants move from the areas of low opportunity to areas of high opportunity (economic motive);
• The choice of destination is regulated by distance;
• Migration takes place in steps, for example, migrants move from rural to nearby towns and then towards large cities;
• Each rural-urban stream produces urban-rural counter stream but former one dominates the latter one.

B. Zipf’s Gravity Model: According to this model, volume of migration of particular stream is inversely related to the distance. In other words people are drawn towards the place of destination by gravitational force which reduces with distance. So the volume of migration at particular place of destination from particular place of origin is directly proportional to the product of the two populations and inversely proportional to the distance between them. So

Where

V= Volume of migration.

Po = Population at place of origin.

P d = Population at place of destination.

This is simple model but fails to explain all type of movements. This does not take into account the characteristics of both the places that could explain volume as well as direction.

C. Lee’s Theory of Migration: Lee’s theory is the general form of Ravenstein law of migration. From this model variety of forces exerting influence can be explained. He terms forces exerting influence on perception of migrants into ‘pluses’ and ‘minuses’ which may be similar to ‘pull’ and ‘push’ factors. In other words, factors which attract individuals towards particular place are pluses or pull factors and those which drive them away are minus or push factors. There are ‘zeros’ also, in which, these forces are, more or less evenly balanced. Lee classified the main factors in the following four groups:

• Factors associated with the area of origin.
• Factors associated with the area of destination.
• Intervening obstacles.
• Personal factors.

D. Todaro’s Model of Rural –Urban Migration: This model has following four major features:

1. Migration is stimulated by rational economic considerations of relative benefits and costs, mostly financial, but also psychological.
2. The decision to migrate depends on expected rather than real Urban-rural wage differentials and the chance of obtaining employment in the urban modern sector.
3. The chance of obtaining an urban job is inversely related to the urban unemployment rate.
4. Migration rates in excess of urban job opportunity growth rates are not only possible but also rational. This may be outcome of continued positive urban-rural expected  income differentials.

The assumptions in Todaro as well as other income differential model are 

1. All potential migrants have equal information about the urban labor market and equal access to urban job.
2. Migrants often look for modern sector job.
3. Wages are lower in traditional sectors compared to modern sectors.
4. Decision to move is a ‘once for all decision.

E. Stouffer’s Model of Migration: Stouffer (1940) stated that the distance is a surrogate for the effect of intervening opportunities. The migration stream effect from origin i to destination j is assumed to be inversely related to the number of intervening opportunities between i and j.  The relevant opportunities may be within a circle centered at i with a radius that equals the distance between i and j. Stouffer 1960 redefined opportunities and included the concept of competing migrants. In this case intervening opportunities fall within a circle centered at mid-point of the distance between i and j with a diameter that equals the distance between i and j. The mathematical form of the intervening opportunities migrant model for interstate migration is given as 

Where 

Y= Gross interstate migration.

XmA =Scale factor measures as the product of total number of in-migrants of j and total number of out migrants from  i during specified period.

XB= Total intervening opportunities measured as total in-migration within a circle whose dia meter is Xo (distance as highway mileage between principal cities of states i and j)

Xo= Total competing migrants measured as total out migration within a circle centered at j whose radius is XD.

K, A, B and C are unknown parameters to be estimated.

F. L-F-R Model of Development: This model was proposed by Lewis 1954 and extended by Rains and Fei (1961). The combined theory is known as L-F-R model. The model is based on a concept of dual economy comprising a subsistence agriculture sector characterized by full employment where “Capitalist” reinvest full amount of their profit. Such migration is equilibrating mechanism which through transfer of labor from the labor-surplus sector to the labor deficit sector brings about the equality between two sectors. These two models also consider the wage differentials in tow sectors as one of the important factors. Their assumption that marginal productivity of labor is zero in subsistence sector however has not been confirmed empirically. Further, migration is not induced solely by unemployment or underemployment.

G. Sjaastad’s Human Investment Theory: This theory treats the decision to migrate as investment decision involving costs and returns distributed over time (Sjaastad 1962). Costs (Money and non-money) include the following components.

1. Cost of transport.
2. Disposal of movable and immovable property.
3. Wages foregone while in transit.
4. Retraining for new job (if necessary).
5. Leaving familiar surroundings.
6. Leaving languages and culture (sometime).
7. Exposure to new dietary habits and social customs.

On the other hand returns may include

• Psychic benefits as change of location (non-money).
• More net life span incomes.

Sjaastad model does not consider the unemployment rate at the place of destination. Further it would be difficult to estimate non money returns and costs in real situation.

H. Cost-Benefit Model: Sjaastad model presented earlier is based on the concept of cost benefit being used by economist in evaluation of family planning program and other development plans. Speare (19:1) proposed a mathematical form of Sjaastad’s model.

Cost benefit model is expressed as:

Where 

Ydj = Earning in the J th year at the destination.

Yaj = Earning in the J th year at the origin.

T = Cost of movement.

N = Total number of years in which returns are expected.

R = Rate of interest used to discount future.

I Mover-Stayer Model of Migration: The mover-stayer model a generalization of the Markov-Chain model, assumes that there are two types of individuals in the population under study:

1. The ‘Stayer’ who remains in the same category with probability one during the study period.
2. The ‘Mover’ whose changes in category over time can be described by a Markov chain with constant transition probability matrix. 

This model was first introduced by Blumen, Kogan and McCarthy (referred as BKM, quoted from Goodman 1961) in their study of the movement of workers among various industrial aggregates in the U.S.

The Markov model requires population homogeneity but transition from origin states rarely conform to this assumption. If the population is heterogeneous in its transition behavior, even if each individual were to satisfy the central assumption of a first order Markov process namely, that his probabilities of making particular transitions are determined solely by his present state and are independent of past history the population level process would not be Markovian. To achieve the homogeneity in the population one may fallow various ways. This may be done by dividing total population into sub-groups considered to be homogeneous. It may also be overcome by assuming that all individuals move according to an identical transition matrix when they move but differ in their rates of mobility. This is the crux of the Mover-Stayer model. It seems that the two categories considered in this model is out of  necessity for keeping the process mathematically tractable not because of population heterogeneity can generally be attributed to two types of persons (Spilerman 1977).

Due to obvious reason we are not giving the detail of this model but interested researchers are advised to read given references. In addition to these references one may also refer some articles of a book edited by Land and Rogers (1982).