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PUSH PULL MIGRATION LAWS x

Guido Dorigo* and Waldo Tobler †

Abstract: The mathematics of a push-pull model are shown to incorporate many of Ravenstein’s

laws of migration, to be equivalent to a quadratic transportation problem, and to be related

to the mathematics of classical continuous flow models. These results yield an improved

class of linear spatial interaction models. Empirical results are presented for one country.

Key Words: geographical movement, Helmholz equation, migration theory, quadratic pro-

gramming, spatial interaction.

It is now approximately one hundred years since the geographer Ernst Ravenstein reported his

“Laws of Migration” to the statisticians of London (Ravenstein 1876, 1885, 1889). We

commemorate this by outlining an elementary mathematical model of migration that incorporates

several of his “laws” as direct and simple consequences. Having studied the literature1, grown

large since Ravenstein’s time, we believe that we can formulate the migration process as the re-

sultant of a “push” factor and a “pull” factor, but must discount this combination by a distance

deterrence between the places. The push factors are those life situations that give one reason to

be dissatisfied with one’s present locale; the pull factors are those attributes of distant places that

make them appear appealing. We will specify this old idea as a very elementary equation system,

and will make estimates using empirical data, but not in a regression format; rather, we will

study the model from a structural point of view. By including distance discounting we place the

model in the venerable class of “gravity” models, and show that it has properties similar to other

models in this class2. But we also show that it is simultaneously an optimizing model, with

shadow prices and with a well-known and simple objective function, and thus belongs to the

class of mathematical programming problems. We then present several computational

algorithms, which give additional insight into the nature of the model. We will interpret the

model in a discrete (network) form, and in a spatially continuous version, the latter as a system

of linear partial differential equations. Our empirical example will use Census Bureau migration

data for the United States.

Mathematical Specification

Ravenstein based his “laws” on insightful, careful scrutiny of census tables. He did not

formulate his ideas in algebraic form, which is what we now attempt to do. Our mathematical

statement is extremely simple, consisting of one elementary equation for each directed exchange

of migrants occurring during a specified interval of time between pairs of places. Yet this simple

system has a deep structure with interesting properties. This is not unusual in scientific

mathematics (May 1976). Specifically, we model migration as

M

ij = (Ri + Ej) / dij , i ≠ j (1)

where Mij is the magnitude (as a count of people) of the movement from place i to place j (of r

places) in some specified time interval, and dij is the distance between these places, measured in

appropriate units (kilometers, road lengths, dollar costs, travel time, social distance, employment

opportunities, etc.). We have labeled our primary variables R and E, using R for “rejecting,”

“repelling,” “repulsing” and E for “enticing.” Ri is the “push” away from place i, and Ej is the

“pull” toward place j. We combine these variables to say that migration is the resultant sum of

the two, discounted for distance effects. This distance effect can be interpreted as an attenuation

of information owing to the two-dimensional geometric nature of the surface of the earth, or as

intervening obstacles to be overcome. Dimensional considerations lead to the conclusion that the

pushes and pulls are in person-kilometers. Thus a push of two hundred will propel two hundred

people one kilometer or one person two hundred kilometers, etc. In our model there are r2 si-

multaneous equations, one for each movement between the pairs of places. When the movements

are between areal units, a distance often is not defined for the self-migrations Mii. Then the

system consists of only r(r-1) equations and the notational convention used here corresponds to

this situation. Self-moves, however, can be incorporated by establishing a rule assigning nonzero

distances for the intraregional moves.

Aggregating the basic Equations (1) over the r places yields

We call these the “outsums” (Oi ) and the “insums” (Ij ).The equations exhibit a certain sym-

metry, so that we can assert that

“the process of dispersion is the inverse of that of absorption, and exhibits similar features”

(Ravenstein 1885, 199).

The important immediate consequence of the aggregation is that it shows that it is pos-

sible to solve for the numerical value of all of the push factors (R) and all of the pull factors (E)

if one knows only the outsums and insums for all of the places (See below for details). These

marginal sums will of course be known if the full migration table is known. In other words, the

equations allow the numerical calculation of the push and pull factors without our knowing in

advance what is decisive for migration. We postulate that the estimated push and pull factors are

combinations (not necessarily linear) of local traits or characteristics of the inhabitants, but we

do not at present need to speculate as to the nature of these attributes of the places or people, and

we do not deny that this is still an oversimplified view of reality. A “true” push factor might be a

high unemployment rate, but this push must be reduced by the heavy inertial cost of leaving

friends and a familiar environment. If we measure the combination of these two tendencies we

may find that the total push at a place is negative. This is not the same as a positive pull toward

that place, but algebraically it does reduce the importance of a pull from some other place. We

will similarly find that an “attractive” place may have a large push value. These results have

been anticipated in a theoretical paper by Lee (1966).

Our model, if valid, has consequences for regression studies. Consider the well known

Lowry (1966) model (Rogers 1968):

Mij =k (Ui Wi / Uj Wj) Li Lj /dij

where the U’s refer to unemployment rates, W’s are wage rates, and the L’s are the number of

people in the respective labor markets. By substitution in (1), and canceling the distance term, we

obtain

Ri – Ej = k (Ui Wj / Uj Wi ) Li Lj ,

which would normally be estimated (with exponents or elasticities) using the logarithmic form.

But the push-pull factors cannot be unscrambled in this way, and the two models are clearly

inconsistent; we will never get good estimates of the push-pull factors from the Lowry model.

But we can run a regression on either R or E, or both, after they have been computed by the

methods outlined below. In this way a correspondence can be established between our model and

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the popular regression models of, for example, Clark and Ballard (1980) or Greenwood (1981).

Once one has calculated the E’s and R’s it is possible to estimate the gross migration

table. That is, one can compute a full movement table from just the marginals and the distances.

In this respect the push-pull model is similar to the origin-destination-constrained entropy model

(Wilson 1967), which is its chief competitor.3 Being able to compute the table from its marginals

is important because this allows a test of the model. We see that the model predictions can be

compared to observed values without any additional calibration of parameters. Empirical results

will be given later. We first deduce more implications of the equation system.

For a place k with a given pull Ek and a hinterland containing places i and j of equal push

Ri and Rj, and with dik < djk , it is immediately apparent that Mik > Mjk. This shows that in these

circumstances migration diminishes in strength with distance, or,

“Migrants enumerated in a ... center of absorption will ... grow less with the distance propor-

tionately" (Ravenstein 1885, 199).

If we let Ak = Ek – Rk then it is easily shown that the net migration flows are Mij - Mji = (Aj -

AI) / dij. In words, the attractivity of a place is the difference between the pull factor and the push

factor at that place. And the net movement between two places is equal to the difference of their

attractivities, discounted by distance, as a gradient. This is the same net migration model used

with some success by Somermeyer (1971) and recently studied in depth by Tobler (1981). It is

seen to be a derivative of the push-pull model.

If we call Tk = Ek + Rk, then the total, two-way, movement between two places is Mij + Mji =

(Ti + Tj) / dij. This ‘T-factor” measures the total exchange, or turnover, at a place; both the in and

out movements are involved. The empirical observations,

“…each main current of migration produces a compensating counter current” (Ravenstein

1885, 199),

and

“inspection of data indicates … outward flows being almost in balance in most cases” (Gleave

and Cordey-Hayes 1977, 17),

are both described by this turnover factor. High values of T imply an active migration market,

with lots of movement in both directions. Low values of T imply a quiescent place, and

intermediate values may mean high in-migration with low out-migration, or the reverse. Some

numerical estimates are given later, as well as a direct computational method by which to

estimate the T values. It would appear that this combination, the sum of the push and pull factors

at a single place, should be an interesting candidate for correlation and regression studies. But it

is not identical with the gross migration sum Mij + Mji or with the total Ik + Ok.

The Equations (2) can be rewritten as

3

This form of the equations shows that the push factor at a place depends on the pull factor of all

of the other places and the number of people leaving the place; similarly, the pull factor at a

place depends on the push factor at all of the other places and the number of people entering the

place. The dependence in both cases is that of a distance decay in a normalized linear

combination, i.e., it is a spatially discounted weighted average. Every place is related to every

other place, but near places are more related, and the push and pull factors are structurally

intertwined. A substantive implication is that if one makes a place less appealing, thus increasing

its “rejectance” or pushing factor, then this will change the enticing pull at all of the other places.

The change in the E’s will then propagate to the alternative places, changing their R values, etc.

Similar ramifications occur if one makes a single place more appealing.

Further, and perhaps less obvious, raising the number of out-migrants (in-migrants) at a place,

for whatever reason or by whatever means, but without changing any other attributes of the

place, increases the push (pull) factors and these changes also propagate through the system.

Notice that these effects are geographically uneven, being more pronounced for outlying places,

and include geographical competition and shadowing. This suggests the following type of

problem (MacKinnon 1975): where should one make a unit change for it to have the greatest

(least) impact on the migration pattern? To the extent that the model describes important prop-

erties of actual behavior, this calculation of impacts of observed or proposed changes at places

on the total migration system can be an extremely useful tool. And, because of the

interrelatedness of push, migration, and pull in the model, there is some prospect that it can be

used for short-range prediction. This is seen by the fact that a change at one place is dampened

spatially relatively quickly (in a short distance; there is no time in the model) and the effect

remains local. The numerical values of most of the R and E factors can therefore be expected to

change only rather slowly with time at most locations. This sketch of model dynamics resembles

relations outlined by Hollingsworth (1970) for migration in Scotland.

It has often been observed that substantial differences in migration rates occur for different

age, occupational, social, and educational classes; Ravenstein was able to distinguish between

male and female migrants. In order to investigate the behavior of different groups let Ma and Mb

be the migration tables for groups a and b. Then the combined table is Ma+b = Ma + Mb and the

components are Ma+bij = (Ra+bi + Ea+bj) / dij, on the assumption that the distances are perceived

similarly for each group. We notice that “exponent additivity” holds for M, R, E, I, O, T and A,

and that the model has this property for any number G of groups,

The push and pull terms are different for each place and each group, as is clear from the notation

used, but we can add these terms to get the same results as would be obtained from an analysis of

the sum of the groups.

Now let each group consist of a single individual. It may be helpful here to imagine the

complete migration table as the sum of a large number of individual migration tables, each

containing only a single entry, with value one, the rest of the entries in that table being zero. Our

empirical example (Table 5) is thus thought of as the sum of some 12 million tables, each of

which describes the movement of one individual. The first Oi of these tables describe the out-

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migrations from region one, and so on. For the kth individual we must have 1 = (Rki + Ekj) / dij or

Rki + Ekj = dij which seems to say that the dissatisfaction at the origin and the expected

satisfaction at a destination for each individual must, at the margin, balance the cost of moving

before the move will occur. Here we have shown that if the push factor is zero then the pull

toward j for the moving individual is proportional to distance. Conversely, if there is no pull then

the push is equal to the cost or distance of moving. In this system each individual is allowed

different satisfactions. Summing over all the individuals who are moving from i to j, we have

which shows that a consistent grouping from individuals to aggregates is possible for this model.

The possibility of such aggregation is generally regarded as being a desirable quality in a

migration model (Speare, Goldstein, and Frey 1974, 163 - 205), and the push-pull model has this

property.

Another interesting computational possibility is to estimate the R and E values for a

single row (or column) from the full migration table. Set all entries to zero except the row

(column) of interest. The computational result is analogous to the estimation of a mean migration

field (Hägerstrand 1957; Tobler 1979). Or divide the row entries by the outsum for that row,

setting all other entries equal to zero (i.e., Mij = Oi (Ri + Ej) / dij ). The migration spread function

that is estimated for the row then bears a resemblance to the impulse response for a unit migrant

(Tobler 1969, 1970).

Variations on the Theme

The extreme simplicity of the model gives it a desirable tinkerability and robustness. For

example, most of the following variations pose no particular problem of calculation or

interpretation. Several of these are of substantive importance.

As noted by Ravenstein (1885. 198),

“In forming an estimate of [the] displacement [of people] we must take into account the

number of natives of each county which furnishes the migrants, as also the population of the …

districts which absorb them”.

Thus some prefer to work with rates and should use Mij = Pi Pj (Ri + Ej) / dij, where the P’s refer

to populations. Adding two such equations, we get Mij + Mji = Pi Pj (Ti + Tj) / dij, which is our

version of Zipf’s (1946) formulation. Subtraction yields Mij - Mji, = Pi Pj (Aj - Ai) / dij, an

attractivity gradient model for net movement rates. The equations for R and E in the above case

are slightly different from those given in (3), but are easily derived by analogous deductions.

Those who prefer different distance-decay functions may use Mij = (Ri + Ej) e-βdij, Mij = (Ri +

Ej) d-αij; or Mij = Bij (Ri + Ej), where Bij is the length of the boundary connecting places i and j.

This latter satisfies the intuitive notion that

“Counties having an extended boundary in proportion to their area, naturally offer greater

facilities for an inflow ... than others with a restricted boundary" (Ravenstein 1885, 175).

5

The crossing density, in persons per linear kilometer, is Mij / Bij = Ri + Ej. This version of the

model allows for easy combination of adjacent regions. Let areas j and k be contiguous to area i.

Then, adding border lengths, we have Bij+k = Bij + Bik . The migration magnitudes are Mij = Bij

(Ri + Ej) and Mik = Bik (Ri + Ek), and, for the combined region, Mij+k = Bij+k (Ri + Ej+k), which

must equal the sum of the first two. A little algebra now demonstrates that the pull factor of the

combined region is the weighted average of the pull factors of the regions that are combined,

Ej+k = (Bij Ej + Bik Ek) / Bij+k

and thus that Mij+k = Bij+kRi + BijEj + BikEk, and this extends to combinations of more than two

regions. Aggregation of regions is thus easily and consistently achieved, and the model does not

require re-estimation. This suggests that the analysis always be performed on the finest level of

geographic detail available.

An Equivalent Model

The model given by (1) is the solution to the variational problem

The equivalence of these two models, (1) and (4), is easily established by writing out the

equations in full (see below). We find that the place-specific push-pull factors in (1) are directly

proportional to the Lagrangians of the constrained optimization problem (4). The quadratic

functional in (4) is the same as the classical definition of the “work” in a resistive electrical

network, and it minimizes this quantity while satisfying the constraints. Zipf’s (1949) postulate

regarding movements is thus equivalent to the postulates of the push-pull model of migration. In

a transportation context it can be shown to be equivalent to minimizing transport costs,

proportional to distance, when congestion causes the costs on each link to increase in direct

proportion to the flow magnitude on that link. T. Smith (in correspondence) has pointed out that

this model also has a simple interpretation as a network equilibrium flow pattern, in the sense of

Wardrop (1952).

An obvious difficulty of the model (1), or of its equivalent (4), is that it is possible to

obtain negative values for the number of people migrating. The condition Mij ≥ 0 must be added.

This can be done in a number of ways, as discussed below, but slightly complicates the

mathematics. Empirically we have discovered that this constraint is not active when the

population-weighted form is used.

Computational Considerations

Equations (2) can be written in matrix form as

where Q is an r-by-r diagonal matrix with

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H is also r-by-r with hij = 1 / dij, i ≠ j, and with a null diagonal. To solve (4) we multiply the

constraints by Lagranians α and β and add them to the objective function, then set the derivatives

equal to zero, in the usual fashion.

Thus the minimum of (4) occurs where the partial derivatives of the constrained objective

function are equal to zero. The first r2 of these derivatives are ∂ε/∂Mij = 2Mij dij - αi - βj. When

these values are zero we have Mij = ½ (αi + βj) / dij which completes the necessary part of the

proof showing the proportional equivalence of (1) and (4). The particular values of αi and βj are

then determined by the partials ∂ε/∂α and ∂ε/∂β. This system of 2r equations is of rank 2r-1,

because

and the R’s and E’s (or α’s and β’s) are determined only up to an additive constant C. Thus the

model can always be written as

Mij =[(Ri + C) + (Ej – C)] / dij.

It is convenient to fix any one R or E and then to solve the reduced set of 2r-1 equations by

taking

The inversion should be calculated using a program for sparse matrices (Jacoby and Kowalik

1980). An alternative approach is to solve Equations (3) directly by a coupled iterative relaxation

technique (Southwell 1956), or to compute a generalized inverse (Bjerhammer 1972). Observe

that the inverse matrix depends only on the geometry of the region; it completely characterizes

the area for this purpose and normally would not differ drastically for sufficiently close time

periods. But this matrix also specifies the interrelationships of all places to each other; any single

change effects everything else in the equations.

One way of approximating the non-negativity condition is to set all negative computed

Mij values to zero and then to use biproportional adjustment (Leontief 1941; Bacharach 1970;

Fienberg 1970) to satisfy the constraints on the marginals. The result is a model of the form

Mij = ai bj (Ri + Ej) / dij,

which, of course, differs from what had been intended. In practice the negative computed

migrations are small, and zeros are forced only for widely separated places, i.e., where small Mij

are expected anyway and the ai bj do not have a large impact on the numerical estimates. The ai

and bj are close to one. The big advantage of this approach is its simplicity, and only a small

amount of computational space is needed. As already noted, the non-negativity constraint is

generally not required when the population-weighted version of the model is used.

The system (4) with the additional requirement Mij ≥ 0 can be solved directly as a quadratic

programming problem with linear constraints (Fletcher 1971). We have used the computer

7

program given in Fletcher (1970). Kunzi, Tzschach, and Zehnder (1971) also give such a

program. The main disadvantage of this approach is the size of the computer memory needed

(about r4), but an exact solution is obtained. Because migration tables on the order of 3,000 by

3,000 are now becoming available, the program cited would need to be run on a computer with

nearly 1014 memory locations, and this is impractical. The sparse structure of the problem

suggests several possible improvements, in addition to the alternatives already cited.

The Equations (3) can be uncoupled as follows. Let T = E + R and A = E - R, and thus,

when dij = dji

where Fk = Ik + Ok, and ∆k = Ik - Ok. Then T and A are seen to be independent of each other, and

each equation can be separately solved by iteration, with ± switches in a single computer

program. Notice the dependence of the turnover T on the sum of the insums and outsums (F) at a

place, and the dependence of the attractivity A on the difference of these values (∆). It is direct

that E = (T + A) / 2 and R = (T - A) / 2. One value is still arbitrary, and care must be taken that

the values satisfy the insum, outsum, and nonnegativity constraints.

The variational form of the model (4) with the non-negativity constraint Mij ≥ 0 suggests

a simpler solution. We can consider this a “quadratic transportation problem,” a generalization of

the well known linear case (Koopmans 1949). The usual tableau is established, but now the

source places are also the destination places, and the tableau is square. The only restriction is that

a place cannot send migrants to itself. Then each place seems to desire people of one kind, and

exports another type, a reasonable enough interpretation. If we can find a simple efficient

algorithm for this quadratic transportation problem, we will not need to use the space consuming

general quadratic programming procedures. An initial feasible solution is easily obtained, and

this is then improved by appropriate changes in the tableau. The constraints are thus always

satisfied. By restricting the changes to integers, and being careful never to subtract more from a

cell than is already there, we move toward a non-negative integral solution. But the true solution

is in general not integer valued (in contrast to the linear transportation problem), so that we must

allow fractional movements in order to achieve the optimum. Philosophically this is slightly

disappointing. Once we have found the optimum, the push, pull factors (Lagrangians) emerge as

shadow prices, that is, as supply and demand prices, like those in the spatial price equilibrium

model of Samuelson (1952), a difference being that in his model linear transportation costs are

minimized. In our spatial equilibrium a quadratic functional is minimized and “location costs”

are assigned to people. The push-pull factors can now be called “prices”; economists often assign

prices to persons, which are then called “labor.” We further observe that the quadratic optimum

contains more than the small number of values that would be obtained from the equivalent (two-

way) linear transportation problem. In fact, the extreme sparcity of entries in the solution to the

conventional linear transportation problem suggests that attempts to model actual, as opposed to

normative, behavior with that model are unrealistic, as has been noted several times (Polenska

1966; Morrill 1967; Mera 1971; Nijkamp 1975). Recently proposals have been made to

overcome this shortcoming, e.g., Hodgson (1978) and Brocker (1980), to which we add our

present suggestion. The advantage of the quadratic functional is that it forces a larger number of

smaller flows, and this is a better descriptor of real tables than is provided by the linear

functional. We are optimistic that the method of Beale (1959) can be improved to obtain a

8

workable primal solution to this nonlinear transportation problem.

After discussing so many computational approaches, we describe a simple algorithm that

appears to work with only modest storage and computation time requirements, and that satisfies

the condition for non-negative movements.4

Step (1): Solve the problem as given by Equation (5) using a matrix inversion, or by operating

on Equations (3) using a coupled convergent iterative scheme.

Step (2): Compute the resulting migrations Mij.

Step (3): Whenever any Mij is negative replace the corresponding dij by a large number (= ∞, 1

/ dij = 0) setting the values in Q and H appropriately, and then go to Step (1). Several of the Mij

may be negative at one time, and all of the appropriate dij are set to large values. Once a dij has

been modified in this manner it is left large on subsequent passes. If there are no negative Mij in

the entire array on this step, then stop because you are done.

The numerical answers obtained by this simple procedure, which seems to require only a

few iterations, agree with those obtained using a full quadratic programming procedure, e.g., that

of Fletcher (1970, 1971).

The Geographically Continuous Version

Ravenstein (1885, 198) asserts that he has

“proved that the great body of our migrants only proceed a short distance,”

and many others have verified this observation (for example, Hägerstrand 1957). Consequently,

it does not do great violence to the data to assume movement only between neighboring places,

especially if observations are taken over very short time intervals. Many types of geographical

movement display this characteristically local process. This is also not as severe an assumption

as might appear at first glance because computationally it simply means that a migrant from, say,

New York to California is routed to pass through all intermediate places.

For expository simplicity allow every place to have only four neighbors among whom ex-

changes are possible. Index these by subscripts from one to four; and take them to be equidistant,

at distance d = 1, and spaced as on a square mesh, for which Bij = 1 also. Then Equations (3)

become

4 E = I - (R1 + A2 + A3 + A4),

4 R = O - (E1 + E2 + E3 + E4).

Now add -4R to both sides of the first equation, and -4E to both sides of the second, to obtain

R

1 + R2+ R3+ R4 - 4 R = l - 4 (E+R).

E

1 + E2 + E3 + E4 - 4 E = O - 4 (E +R).

The left hand sides are recognized as finite difference versions of the Laplacian. Thus we can

write, approximately and for a limiting fine mesh,

Λ2 E(x,y) = I(x,y) - 4 [R(x,y) + E(x,y)],

Λ2 R(x,y) = O(x.y) - 4 [R(x,y) + E(x,y)],

assuming that R and E are differentiable spatial functions5 . This is a coupled system of two

simultaneous partial differential (Helmholtz) equations, each of which separately is the Euler

equation of, or makes stationary, a functional of the form

9

with ∂z/∂n = 0 on the boundary. The equations can be uncoupled as before. By addition we

obtain

Λ2 T + 8 T = F,

where F(x,y) = I(x,y) + O(x,y) is the forcing function. This is a single Helmhotz equation in the

single unknown turnover function T(x,y), minimizing an integral of the given type. Alternatively,

by subtraction we have

Λ2 A = ∆

using ∆(x,y) = I(x,y) - O(x,y); this result gives Poisson’s equation for the unknown A(x,y).

Efficient computer algorithms exist for solving both Helmholz’s equation and Poisson’s equation

(Proskurowski and Widlund 1976; Buzbee, Golub, and Nielson 1970). The arbitrary additive

constant of integration must still be supplied. A(x,y) is interpreted as the attractivity at each

location. Assuming that movement is proportional to the gradient of this attractivity, i.e., V =

grad A, allows one to make interesting maps of the net geographical movement pattern, as is

shown in a later figure. The movement across the border surrounding the region being studied

gives the Neumann condition for these partial differential equations. In Table 5 (below) this flux

is not recorded and is treated as implying no international migration into or out of the United

States.

The occurrence of Helmholz’s equation, normally used to describe dynamic wave mo-

tion, in a static equilibrium paper on migration, was quite unanticipated. Hydrologic terminology

(streams, currents, flows, eddies, waves) is common in the migration literature (e.g., Redford

1964), but the equations are not.

Empirical Example

We have performed a few computations contrasting the push, pull model with US Bureau

of the Census (1973) migration estimates for 1965-1970. These estimates are based on a 15

percent sample enumeration of the migrations between the nine census regions defined by the

government for data publication purposes. Nine regions covering the contiguous United States

yield an average geographic resolution of 944 kilometers, and patterns of circa 2,000 kilometers

across or greater should be observable. Tables 1 and 2 give input data for the model, including

highway driving distances between cities located near the center of gravity of the regions and the

length of the boundaries between the regions, the latter computed from a magnetic tape

containing the boundaries of all of the counties of the United States in coordinate (latitude,

longitude) form. Table 3 presents our estimates for the basic push-pull model and has been

computed by the constrained quadratic programming algorithm. Whenever a negative flow is

computed from the push-pull factors of this table, it must be set to zero to obtain the correct

minimum of the functional. The fit of the model and its variants to the data can be measured in

several ways; all indicate about the same results as are usually obtained for gravity models (R2 >

80 percent). More interesting are the maps (Figure 1), which show the resulting push, pull,

turnover, and attractivity values in their geographical context. The Pacific Region is seen to be

the most “repulsive.” High housing costs or metropolitan air pollution might be the reason.

Conversely, this same area is also the most “enticing,” perhaps for its beautiful scenery, coastal

climate, or life styles. Our model does not consider these contrasts contradictory but instead

incorporates, by summation, these two effects into a large turnover, and also assigns a high net

attractivity to the region. One may compare regions known to be losing population, the Mid-

Atlantic for example, in this same way. This area still has a large drawing power, but this is

10

overcome by the push effect. Each region on the maps can be examined in this manner.

Figure 2 shows the comparable results for the model

Mij = Pi Pj (Ri + Ej) / dij.

The effect of removing the population sizes is seen. Here is a case in which the matrix

formulation (Equation 5, as modified to include population) was used directly and the Mij ≥ 0

constraint was not needed; all values turned out to be positive even without the use of this

constraint. Table 4 shows the estimated flows, and Table 5 repeats the census figures for easy

direct comparison.

Tables 6 and 7 show the solution using

M

ij = Bij (Ri + Ej) (6)

with the lengths of boundaries between regions instead of distances. This result is particularly

interesting because we have had to introduce pass-through transients in order that all of the flows

come out correctly. It is not possible to get all of the necessary people to the Pacific Region from

the Mountain Region alone. Nor can this latter region absorb all of the out-migrants from the

Pacific Region. We must increase the insum to the Mountain Region, and increase its outsum

also, to get enough people to move from the East to the West, and in the opposite direction. The

following algorithm seems to work well. Compute a solution using Equations (5), appropriately

modified to use boundary lengths instead of distances. Negative migration flows will occur when

this solution is inserted into the basic model (6). Now increment the insums and outsums by the

absolute value of the sum of the negative computed migrations in each row or column. Use these

new marginals to get a new estimate via (5) - the inverse depends only on the geometric structure

of the region and need not be recomputed. Continue this iterative procedure until there are no

negative flows. This appears to be a new way of solving the “traffic assignment problem”

(Florian 1976; Boyce 1980), routing all flows through adjacent places, while minimizing

congestion via the quadratic functional.6 Table 7 gives the number of estimated border-crossing

transients by region. These transshipment migrations

“sweep along with them many of the natives of the counties through which they pass . . . [and]

deposit, in their progress, many of the migrants which have joined them at their origin”

(Ravenstein 1885, 191).

Unfortunately we know of no actual published counts on the part of statistical agencies of

crossings of internal boundaries by migrants. International estimates may be easier to obtain.

Until such data are found there is no method of testing the realism of our procedure. But it is

easily seen that

“…even in the case of ‘counties of dispersion’, which have population to spare for other coun-

ties, there takes place an inflow of migrants across that border which lies furthest away from the

great centers of absorption”. (Ravenstein 1885, 191).

In the present instance the West North Central states form one such interior lying dispersing

region; the reader will be able to find others and can easily identify the direction of movement

across each of their borders. A striking contemporary international case is Mexico, which has an

illegal immigrant problem - with people coming in from the south of the country!

The final maps (Figure 3) display the results of a computation based on the continuous

attractivity model using Poisson’s equation

Λ2 A(x,y) = I(x,y) - O(x,y).

The contiguous United States is approximated, somewhat crudely, by a 61-by-95 lattice of 5,795

nodes, with one finite difference equation for each node. The changes in population resulting

from migration are distributed over each of the nine census regions to yield the source/sink field

11

∆(x,y) on this mesh. Computation of the potential A(x,y) and the flow vectors then uses standard

procedures (Wachspress 1966) with a Neumann condition on the boundary.7 The continuous case

closely resembles the case in which transients cross over internal borders, and it is in fact

possible to count the number of transients in this model; the method is described in detail in

Tobler (1981). The resulting maps indicate that

“…migratory currents flow along certain well defined geographical channels” (Ravenstein

1889: 284),

although this shows up much better when higher-resolution data are used as input or when time

series are available. The variation in the density of the streaklines, and the spacing of the

contoured potentials, also allows one to see that

“The more distance from the fountainhead which feeds them, the less swiftly do these currents

flow”. (Ravenstein 1885: 191).

The streaklines should be interpreted as ensemble averages rather than as paths of individual

migrants. It is also apparent that no account has been taken of transportation facilities in the

computation for Figure 3. The figure does not show fine detail, such as movement to the suburbs,

because of the low resolution of the migration observations.

Conclusion

We believe that we have adequately demonstrated that several of Ravenstein’s “laws” hold for

our equations. There remain many challenging problems, as should be obvious from all of those

aspects of migration that we have not modeled, and we are aware of these inadequacies. Still, as

the Nobel laureate Paul Samuelson so aptly remarks in his Economics (1976):

“Every theory, whether in the physical or biological or social sciences, distorts reality in that

it oversimplifies. But if it is a good theory, what is omitted is outweighed by the beam of

illumination and understanding thrown over the diverse empirical data".

Our model obviously does not fit the observations perfectly; perhaps the residuals will suggest

alternative research directions. But we believe that some issues have been “illuminated.” We

have given a specific mathematical form, previously lacking, to the push-pull idea and have

shown that many of Ravenstein’s “laws” can be deduced from these formulae. We have

established relations to Zipf’s “principle” and to conventional spatial interaction models, and

have shown how these can be contrasted with the now classical transportation/transshipment

problem and the spatial price equilibrium concept. A spatially continuous version of our model is

articulated; this should become increasingly important as migration tables increase in size to

provide greater spatial resolution. Our model yields two-way migrations within the context of

pushes and pulls and is consistent at individual and aggregate levels; both attributes have been

difficult of specification in previous work. The astute reader will also have observed that we

have described an abstract geographical movement system and not only, or even specifically,

migration. The model equations can therefore be applied to information flows, to commodity

movements, to commuting patterns, or to shopping behavior, and so on. The model may also

have relevance to sociometric matrices (Holland and Leinhardt 1981).

We have specified several model formulations and must now discriminate between these

variants and between other models in the literature, especially those incorporating the entropy

approach introduced by A. Wilson (1967). Some recent suggestions have been made for this

12

purpose (Bishop, Fienberg, and Holland 1975, Chap. 9; Kau and Sirmans 1979; Hubert and

Golledge 1981) but need refinement because several of our proposed variants fit so well that they

cannot be distinguished statistically on the basis of the data used here. The linear property of our

model gives it mathematical advantages over the multiplicative entropy class, especially with

respect to aggregation. No analogy to physics has been used in the development of our model,

but the resulting equations clearly bear a resemblance to those encountered in continuum

mechanics. The entropy equations are derived from ideas also used in statistical mechanics. Thus

the two classes of models of spatial interaction may bear the same relation to each other as the

two approaches to mechanics. Each approach attempts to describe the same events as the other

and uses comparable empirical observations. As outlined here the push-pull model is

deterministic, but a stochastic version may also be obtainable (Hersh and Griego 1969). In-

vestigation of the relation between these classes of models may lead to deeper insights, and this

offers a challenge for students.

Computationally. we have assumed either the availability of the migrant insums and out-

sums (or their combination I + O), or the equivalent availability of place-specific person prices

(push-pull factors). Much of the migration literature is concerned with explaining these factors.

We have shown how to compute them a posteriori. This has advantages and disadvantages,

Instead of attempting to include complex human behavior in our equations, we estimate the

effects of this behavior, allowing other researchers to estimate the relation of the computed

pushes and pulls to life situations. As a consequence we cannot claim to have explained why

people migrate; we only assert a tolerably good description of the spatial pattern when one is

given table marginals and a measure of the strength of driving impulses, whatever these might

be. Because of this, our model is equally applicable to refugees, who constitute a large fraction

of all migrants, as well as to the more benign internal movements. A remaining difficulty is in

the dynamic forecasting of the insums and outsums, the feedback these have on subsequent

events, and in the details of the temporal and spatial lag structures.

Acknowledgments

Dr. Dorigo’s postdoctoral studies in the United States were supported by the Swiss National

Fund for Scientific Research. We thank Teresa Everett, Susan Pond, and Joanne Mustacchia for

their assistance in completing the manuscript and Ralph Milliff for computation of the flow

maps.

Notes

1. The reader is directed to the several bibliographic and critical surveys that have recently been

published, e.g., Adams (1968), Bennett and Gade (1979), Clark (1982), Cordey-Hayes

(1972), Courgeau (1980), DaVanzo (1981), Davis (1974), Golledge (1980), Greenwood

(1975), l-(âgerstrand (1957), Hoffmann-NoVotny (1970). Hotelling (1978), Kuznets (1957-

1964), Ledent (1981), Mangalam (1968), Morrison (1973), Mueller (1982), Olssori (1965),

Petersen (1978), Petersen and Thomas (1968), Pierson (1973), Price and Sikes (1975), Pryor

(1981), Quigley and Weinberg (1977), Ritchey (1976), Shaw (1975), Thomas (1938),

Thornthwaite (1934), Weaver and Downing (1976), Welch (1971), Willis (1974), Wolpert

(1965), the International Migration Reviaw, a periodical published by the Center for

Migration Studies of New York, and the Population Index of Princeton University.

2. Stewart 1947; Carrothers 1956; Ewing 1981; Schneider 1959; lsard 1960; Batty 1976;

Smith 1978; Tocalis 1979; Hua and Porell 1979; Erlander 1980.

13

3. Wilson’s model is multiplicative Mij =ai bj Oi Ij exp( -β dij) and maximizes a constrained

entropy function. An additive analog with the same constraints is

Here D is a constraint on the total movement, and this leads to the slightly different push - pull,

distance-deterence model: Mij = Ri + Ej + γ dij. Empirically, using the nine-region census data

(Tables 2 and 5), we find that γ is negative, as expected. The computational procedures are

similar to those described for the model (1); also see Wansbeek (1977). We deduce for this

model that Mij + Mji = Ti + Tj + 2 γ dij and Mij – Mji = Ai – Aj when dij= dji. (T and A are as

defined in the body of the text.) Mii is also estimated if included with the in-and outsums.

4. The computer program is available from the authors.

5. We use the standard notation

for the Laplacian operator.

6. On a network one might also wish to add capacity constraints.

7. The boundary condition is only approximately satisfied by our numerical technique; see

Milliff (1980) for a discussion of this problem.

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18

Table 1: Model Inputs

Region Insum Outsum I + 0 I - 0 1970

Population

New England 675,408 679,180 1.354,588 —3,772 11,848,000

Mid Atlantic 1,155,811 1,874,320 3,030,131 —718,509 37,056,000

E North Central 1,789,112 2,134,267 3,923,379 —345,155 40,266,000

W North Central 942,162 1,212,105 2,154,267 —269,943 16,327,000

South Atlantic 2,484,387 1,765,650 4,250,007 718,737 29,920,000

E South Central 819,222 986,050 1,805,272 —166,828 13,096,000

W South Central 1,237,079 1.146,498 2,383,577 90,581 19,025,000

Mountain 1,067,069 987,331 2,054,400 79,738 8.289,000

Pacific 2,143,172 1,528,021 3,671,193 615,151 25.476,000

Source: U.S. Bureau of the Census (1973).

Table 2: Model Inputs

Distance between cities below the diagonal.

Length of border between regions above the diagonal.

All in miles.

1 2 3 4 5 6 7 8 9

1 NE: Boston — 301 0 0 0 0 0 0 0

2 MA: New York 219 — 91 0 350 0 0 0 0

3 ENC: Chicago 1,009 831 — 972 264 703 0 0 0

4 WNC: Omaha 1,514 1,336 505 — 0 166 755 937 0

5 SA: Charleston 974 755 1,019 1,370 — 1.295 0 0 0

6 ESC: Birmingham 1,268 1,049 662 888 482 — 1,140 0 0

7 WSC: Dallas 1,795 1,576 933 654 1,144 662 — 637 0

8 MTN: S Lake City 2.420 2,242 1,451 946 2,278 1,795 1,287 — 154

2

9 PAC: S Francisco 3,174 2,996 2,205 1.700 2,862 2,380 1,779 754 —

19

20

Table 7. Model Results. Estimated from Mij = Bij(Ri + Ej)

Push Pull Turnover Attractivity

Region (R) (E) (T= E + A) ( A = E - R)

Transients

New England 2,969 0 2,969 —2.969 0

Mid Atlantic 2,240 —715 1,525 —2.955 0

E North Central 1,440 —74 1,366 —1,514 13,611

W North Central 1,210 —24 1,186 —1,234 1,368,074

S Atlantic 1,886 618 2,504 —1,268 0

E South Central 106 —1,210 —1,104 —1,316 231,004

W South Central 1,221 106 1.327 —1,115 95,387

Mountain 43 —707 —664 —750 1,269,506

Pacific 1,698 1,347 3,045 —351 0

Figure 1. Push, pull, turnover, and attractivity values from the basic model (Table 3).

21

Figure 2. Push, pull, turnover, and attractivity values from the Mij = PiPj(Ri + Ej)/dlj model

Figure 3. Potential field, gradients, and streaklines for 1965/1970 from the model

Λ2 A = ∆, with ∂A/∂n = 0 on the boundary, solved as a system of 5,795 simultaneous equations.

22