Chapter 4
Newton’s First Law of Migration: The Gravity Model
A. Logistics
Student’s Time Requirements
Activity 1: Predicting Migration with the Gravity Model 15–30 minutes
Activity 2: Scatter Diagram 10–20 minutes
Activity 3: Residual Map 15–25 minutes
Activity 4: Evaluation 40–60 minutes
We recommend you do not skip any parts of Activity 1. The material is cumulative in the sense that Activity 2 builds upon Activity 1, Activity 3 builds upon Activities 1 and 2, and so on. Activity 1 involves simple spreadsheet mechanics, but it provides the raw material for the critical thinking that comes in analyzing the scatter diagram in Activity 2 and analyzing residuals in Activity 3. Activities 1–3 are necessary for evaluating the ability of the gravity model to predict migration flows and for suggesting revisions to the model that enhance its predictive ability in Activity 4. You may wish to tell students to look at the questions in Activity 4 before they complete and exit out of Activities 1–3. This will help them think about what they are doing in the first three activities.
Activities 1–3 use access to the Internet. This can be done individually or in groups, in a computer lab or outside of class. In using the Internet, students gain experience interrelating spatial information from a spreadsheet (Activity 1), a scatter diagram (Activity 2), and a residual map (Activity 3). We recommend that you demonstrate Activities 1–3 using a computer projection system if one is available. You can do so without giving away the answers by demonstrating the functionality with a different state. For example, if you are in California, try using Oregon or Florida as your example.
In Activity 4, students respond to a series of open-ended questions about the gravity model’s ability to predict migration to their state. For most U.S. states and Canadian provinces, the gravity model is only a mediocre predictor of migration. The fun and challenge of the activity is in identifying the outliers—the states and provinces that are poorly predicted—and in figuring out how the model can be modified so it works better. Think of the gravity model as a straw man: We are trying to knock it down and pick it apart. It helps to emphasize that the gravity model reveals something, but not everything, about migration.
You may wish to preface this exercise with a discussion of migration to your state or province:
· Why might some states send more migrants to your state than others?
· What do you think will be the major sending areas for migrants to your state? Why?
B. Lesson Plan
I. Places are connected through spatial interaction. Interactions can be movements of:
1. Ideas
2. Information
3. Money
4. Products
5. People
II. Migration
1. Defined as permanent change in residence to outside one’s community of origin
2. Spatial scale and migration
3. Factors of place desirability
4. Immigration as special case of migration across international borders
a. Geography of immigrant sending regions through time
b. Remittances
c. Global immigration patterns
d. Decline in undocumented Mexican migration to the United States
e. Integration of U.S. and Mexican economies and societies
f. Refugees as special case of immigrant—threat of persecution
5. Push and pull factors
6. Migration selectivity factors (age, education, length of residence)
7. Distance decay
8. Migration streams and counterstreams
a. Examples: Dustbowl migrants, immigrants to enclaves, Cuban- and Mexican-born immigrants
III. The Gravity Model
1. What is a model? A simplified representation of reality
2. The gravity model was adapted from the physical sciences where it is used to show the attraction between two masses
3. The gravity model was adapted to the social sciences to show spatial interaction between two places: William J. Reilly in 1931
4. Mathematical formulation
a. Size or mass variables in the numerator
b. Distance variable in the denominator
c. Distance exponent—measures the drag, or impedance, of distance on interaction
d. K factor
5. Think critically about modeling human behavior
IV. Mobility
1. Part of the American experience
2. High in developed countries with immigrant background
3. Strong predictor of whether person will move in the future is whether they have moved in the past
4. Currently a lower rate of U.S. mobility than in earlier decades
5. Regional differences in mobility rates
6. Regional and sub-regional shifts in population
7. Net migration
8. Today’s migration patterns reflect:
a. The location of states relative to one another (nearby states tend to exchange migrants)
b. Historical patterns of movement (i.e., longtime linkages between Florida and New York and between California and Texas)
c. The changing geography of economic opportunity in the nation
d. The public’s perceptions about the attractiveness of places, including intangibles such as an agreeable climate, being near family and friends, and a good view
V. Review the mechanics of the assignment in Activities 1–3.
1. Spreadsheets allow easy calculation of an entire series of numbers in a few simple steps; scatter diagrams allow us to plot actual versus expected migration.
2. Outliers are states that deviate from the 45-degree line on the scatter diagram.
3. Residuals show errors in the predicted values.
VI. Discuss Activities 1–3.
C. Answer Key
Activity 1: Predicting Migration with the Gravity Model
Spreadsheet results from Activity 1 will vary from state to state (or province) because you are using different migration data and different k coefficients. You will need to verify that students have followed the correct formula for the gravity model for their particular state. The scatter diagram in Activity 2 and the residual map in Activity 3 are programmed by the software to be correct.
Activity 2: Scatter Diagram
The graph to be handed in will show actual migration on the x-axis versus predicted migration on the y-axis. You should look for two features on these graphs because students have been asked to customize their graphs in two ways. First, the students have been instructed to delete extreme values (very high x and/or very high y values) so that the majority of points will be spread out along the diagonal line. Students who fail to delete extreme values will find that most of their points are clustered into the bottom left corner, making interpretation difficult. Second, up to five outlier states should be labeled (by hand if turning in hard copies). Outliers are poorly-predicted states, that is, dots that do not line up near the 45-degree diagonal line.
Activity 3: Residual Map
In Activity 3 students are asked to customize their residual maps by choosing reasonable class limits and a good color scheme. One approach is to have an even number of classes with a break point at zero. An alternative approach is to choose an odd number of classes, with the middle class symmetrical around near-zero residuals. The color/shading scheme should clearly distinguish between positive and negative residuals and between low and high residuals. Color schemes that look good and clearly differentiate between positive and negative residuals on the screen may be a poor choice when printing a black-and-white map.
Activity 4: Evaluation
4.1 Students must make clear which states or provinces, if any, they eliminated from the scatterplot. They must also justify their residual class breakpoints. Be sure they clearly show differences between positive and negative residuals (the red/blue combination color scheme works best for this), and that the most poorly-predicted states or provinces above and below the best-fit line show on the map.
4.2 The assessment of how well the gravity model predicts migration should be based on (a) how well the points on the graph line up along the diagonal, or (b) the typical size of the percentage residuals. For most states there is a rough correspondence between the actual and predicted migration indicating that the gravity model provides at least a preliminary picture of the major sending areas for migration. In general, the line of prediction fits the trend of the scatter. Some students will try to quantify their answers, for example, stating that 75 percent of the states were predicted within 500 migrants, or predictions for 80 percent of the states were within 20 percent of the actual migration. Refer students to Figure 4.14 that shows how error radiates out from the origin, so the error distance is relative to the location on the graph.
Note: Many Canadian provinces have unusual-looking scatter diagrams, in which only one or two provinces are on one side of the line and all the others are on the other side of the line and the line falls in between. It is reasonable for students in such cases to say the model does not seem to fit well at all. Canada has a small sample size, and is dominated by two large provinces: Ontario and Quebec. The k coefficients were estimated by a least-squares linear regression of Mij on Pi/dij with the constant term forced to zero (that is, using the functional form Y = kX, where Y = Mij and X = Pi/dij). The square of Ontario’s or Quebec’s residual was often much larger than any (or even all) of the other provinces. Thus, the line-of-best-fit is pulled towards the Ontario and/or Quebec observation, often leaving the line a poor visual fit to any one group of observations.
4.3 Distance acts as a deterrent to migration in several ways: cost, information, and intervening opportunities. The farther apart two places are, the more expensive it is to move between them. Also, people are unlikely to move to a place they know nothing about, and they tend to know less about distant places than about those nearby places. And finally, the farther apart two places are, the greater the likelihood there will be viable options in between—the idea of intervening opportunities. The time it takes to complete a move should probably not be mentioned as a factor that discourages long-distance moves. Students who mention that are probably making an incorrect analogy to daily commuting and much more frequent forms of spatial interaction.
4.4 Population is justified in the gravity model because the more people live in a state, the more potential migrants there are. If distance were held constant, one would expect migration to be proportional to the population of a state. However, a better measure of the “sending power” of a state would take into account not only how many people live in a state, but also what kinds of people are most likely to move. Therefore, there might be a higher weighting on young adults and highly-educated people who move more frequently than others (see Figure 4.4 on age-specific mobility rates to get students to think about this answer).
4.5 For points below the line, actual migration is greater than predicted migration, the gravity model predictions underestimate migration, and residuals (actual–predicted migration) are positive. For points above the line, predicted migration is larger than actual migration, the gravity model predictions overestimate migration and residuals are negative. Figure 4.13 gives students the answer.
4.6 This answer will vary from state to state (or province). However, a few generalizations can be offered. First, students should focus on regional groupings of states with similar residuals rather than on single states. For example, they may note that nearby states tend to have large positive residuals or that southern states have large negative residuals. Second, they should attempt to explain why these regional groupings have high positive or negative residuals. Keep in mind that a large positive residual means that actual migration is greater than what would be expected on the basis of population and distance alone. States may send more migrants than expected because of economic considerations (high unemployment rates, high housing costs, low incomes), climatic factors (too cold in the winter or too hot in the summer), or cultural factors (the state is similar in religion, foods, dress, and other lifestyle preferences). Similarly, states with large negative residuals send fewer migrants than expected given their populations and distances. These states may have particularly good economic or climate conditions, have dissimilar cultures, or send their migrants to different destinations.
Note: Many states appear to have positive residuals in the western area of the country. We spent a lot of time checking this to see if any systematic errors or biases were behind it, and we did not find anything. Our best explanation for this is simply that mobility rates are higher in the West, and that there is a lot of return migration from the West to other states.
Note: Students sometimes notice that nearby states have positive residuals and faraway states have negative ones. This arises because we have not specified a distance exponent and have therefore assumed it to be 1.0. An increase in the distance exponent would increase the drag of distance on spatial interaction; it could make the distance decay curve steeper. This would have the effect of increasing migration from nearby states and decreasing migration from distant states. Some very bright students will see this from the residual map and suggest that the distance exponent in the gravity model be increased to better represent migration to their state.