Gravity Models of Trade and Migration
modelling Level 3

Gravity Models of Trade and Migration

People interact more with nearby places than distant ones. Trade, migration, communication — all decay with distance. The gravity model captures this using an inverse power law borrowed from physics. This model shows how mathematical patterns cross disciplinary boundaries.

Prerequisites: inverse power laws, distance decay, parameter estimation, sensitivity analysis

Updated 11 min read

1. The Question

Why do cities trade more with nearby cities than distant ones?

A person in Denver is more likely to:

  • Shop in Boulder (40 km away) than in Phoenix (900 km)
  • Migrate to Fort Collins (100 km) than to Seattle (2000 km)
  • Commute to Aurora (20 km) than to Colorado Springs (110 km)

Distance matters. But how exactly?

The mathematical question: Can we write down a formula that predicts the strength of interaction between two places based on their sizes and the distance separating them?

2. The Conceptual Model

The Gravity Analogy

In physics, Newton’s law of gravitation states:

\[F = G \frac{m_1 m_2}{r^2}\]

Gravitational force is:

  • Proportional to the masses of the two objects
  • Inversely proportional to the square of the distance between them

Human geography insight: Maybe social and economic interactions follow a similar pattern?

Gravity model of spatial interaction:

\[I_{ij} = k \frac{P_i^\alpha P_j^\beta}{d_{ij}^\gamma}\]

Where:

  • $I_{ij}$ is the interaction between place $i$ and place $j$ (migration flow, trade volume, phone calls, etc.)
  • $P_i, P_j$ are the populations (or economic sizes) of the two places
  • $d_{ij}$ is the distance between them
  • $k$ is a constant (scaling factor)
  • $\alpha, \beta, \gamma$ are exponents (often $\alpha = \beta = 1$, $\gamma \approx 1$–$2$)

Distance Decay

The term $d_{ij}^{-\gamma}$ is called distance decay.

As distance increases, interaction decreases rapidly:

  • If $\gamma = 1$: doubling distance halves interaction
  • If $\gamma = 2$: doubling distance quarters interaction (like gravity)
  • If $\gamma = 3$: doubling distance reduces interaction to 1/8

Key insight: The exponent $\gamma$ determines how sensitive interaction is to distance.

3. Building the Mathematical Model

Simplest Form

Assume populations enter symmetrically ($\alpha = \beta = 1$) and distance decay follows an inverse square law ($\gamma = 2$):

\[I_{ij} = k \frac{P_i P_j}{d_{ij}^2}\]

Interpretation:

  • Numerator: Larger populations → more interaction (more potential traders, migrants, commuters)
  • Denominator: Greater distance → less interaction (higher travel cost, time, uncertainty)

Calibrating the Exponent

Real data rarely fit $\gamma = 2$ exactly. The exponent depends on:

  • Type of interaction: Commuting ($\gamma \approx 1.5$), migration ($\gamma \approx 1$–$2$), trade ($\gamma \approx 1.5$–$2.5$)
  • Transportation technology: Better roads, cheaper flights → lower $\gamma$ (distance matters less)
  • Cultural/linguistic barriers: Higher $\gamma$ across language boundaries

Estimating $\gamma$: Fit the model to observed data using regression.

Take logarithms:

\[\ln I_{ij} = \ln k + \ln P_i + \ln P_j - \gamma \ln d_{ij}\]

This is linear in the logs. Regress $\ln I_{ij}$ on $\ln P_i$, $\ln P_j$, and $\ln d_{ij}$ to estimate $\gamma$.

Competing Destinations

If person in city $i$ is choosing where to migrate among multiple destinations, the probability of choosing destination $j$ is:

\[\text{Prob}(j | i) = \frac{P_j / d_{ij}^\gamma}{\sum_{k} P_k / d_{ik}^\gamma}\]

This is a discrete choice model — the denominator normalizes so probabilities sum to 1.

Interpretation: Larger, closer cities are more attractive destinations.

4. Worked Example by Hand

Problem: Three cities with populations and distances:

City Population Distance from A
A 100,000 0
B 50,000 100 km
C 200,000 200 km

Using the gravity model $I_{ij} = k \frac{P_i P_j}{d_{ij}^2}$ with $k = 1$:

(a) What is the interaction $I_{AB}$ (A ↔ B)?
(b) What is the interaction $I_{AC}$ (A ↔ C)?
(c) Which city does A interact more with?

Solution

(a) Interaction A ↔ B

\[I_{AB} = \frac{100000 \times 50000}{100^2} = \frac{5 \times 10^9}{10000} = 500000\]

(b) Interaction A ↔ C

\[I_{AC} = \frac{100000 \times 200000}{200^2} = \frac{2 \times 10^{10}}{40000} = 500000\]

(c) Comparison

$I_{AB} = I_{AC} = 500000$

A interacts equally with B and C.

Why? City C is twice as far (200 km vs. 100 km) but has four times the population (200,000 vs. 50,000). With $\gamma = 2$, these effects exactly cancel:

\[\frac{200000}{200^2} = \frac{50000}{100^2}\]

If $\gamma$ were larger (stronger distance decay), A would interact more with B. If $\gamma$ were smaller (weaker distance decay), A would interact more with C.

5. Computational Implementation

Below is an interactive gravity model with adjustable parameters.

Try this:

  • Set $\gamma = 0.5$ (weak distance decay): Large distant cities dominate
  • Set $\gamma = 3.0$ (strong distance decay): Only nearby cities matter
  • Change origin city: See how interaction patterns shift
  • Increase number of cities: More competing destinations

Key insight: The exponent $\gamma$ is a behavioral parameter — it controls how much people care about distance vs. destination size.

6. Interpretation

Why Distance Decay Exists

Costs increase with distance:

  • Time: Longer travel takes more hours (lost productivity, fatigue)
  • Money: Fuel, tickets, shipping fees scale with distance
  • Uncertainty: Information about distant places is less reliable
  • Social ties: Harder to maintain relationships across long distances

Result: The friction of distance reduces interaction.

Cultural and Technological Shifts

Historical change:

  • 1800s: $\gamma \approx 3$–$4$ (horse-drawn transport, poor roads)
  • 1950s: $\gamma \approx 2$ (cars, highways, telephones)
  • 2020s: $\gamma \approx 1$–$1.5$ (air travel, internet, remote work)

Interpretation: Globalization = declining $\gamma$.

Borders and Barriers

The gravity model often underpredicts cross-border flows because:

  • Language differences
  • Currency and trade regulations
  • Cultural distance (not just spatial distance)

Modified model: Add a border effect term:

\[I_{ij} = k \frac{P_i P_j}{d_{ij}^\gamma} \times e^{-\delta B_{ij}}\]

Where $B_{ij} = 1$ if $i$ and $j$ are in different countries, 0 otherwise.

7. What Could Go Wrong?

Assuming Symmetry

The model assumes $I_{ij} = I_{ji}$ (symmetric flows). In reality:

  • Migration: More people move from rural to urban areas than the reverse
  • Trade: Exports from A to B ≠ imports from B to A

Solution: Use origin-constrained or destination-constrained models that fix row or column totals.

Measuring Distance

Euclidean distance (straight-line) is simple but unrealistic.

Alternatives:

  • Road network distance (actual travel routes)
  • Travel time (accounts for speed limits, congestion)
  • Economic distance (cost of transport)

For air travel, straight-line distance is reasonable. For road freight, network distance matters.

Omitting Intervening Opportunities

If city C lies between A and B, people from A may stop in C rather than continuing to B — even if B is larger.

The gravity model doesn’t capture this spatial competition effect.

Solution: Intervening opportunities model — interaction decreases with the number of opportunities passed en route.

Aggregation Bias

The model uses city-level populations. In reality, individuals vary:

  • Some people are highly mobile (frequent travelers)
  • Others are tied to place (elderly, families, low income)

Aggregate models obscure individual heterogeneity.

8. Extension: Retail Gravity Models

A shopper choosing between two stores follows the same logic:

\[\text{Prob}(\text{Store } j) = \frac{S_j / d_j^\gamma}{\sum_k S_k / d_k^\gamma}\]

Where $S_j$ is the size (floor area, product variety) of store $j$.

Application: Retailers use this to predict market share and choose store locations.

Reilly’s Law of Retail Gravitation (1931): The breaking point between two competing cities A and B is:

\[d_A = \frac{d_{AB}}{1 + \sqrt{P_B / P_A}}\]

Where $d_{AB}$ is the distance between A and B. Customers closer than $d_A$ to city A will shop in A; those farther will shop in B.

9. Math Refresher: Power Laws vs. Exponentials

Power Law

\[y = ax^{-\gamma}\]
  • Straight line on log-log plot: $\ln y = \ln a - \gamma \ln x$
  • No characteristic scale — the ratio $y(2x) / y(x)$ is constant regardless of $x$
  • Heavy tails — large values occur more often than in exponential distributions

Examples: Gravity model, city size distributions (Zipf’s law), earthquake magnitudes

Exponential Decay

\[y = ae^{-\lambda x}\]
  • Straight line on semi-log plot: $\ln y = \ln a - \lambda x$
  • Characteristic scale: $1/\lambda$ is the distance where $y$ drops to $1/e \approx 37\%$
  • Light tails — extreme values are rare

Examples: Radioactive decay, light attenuation, population density with distance from city center

Which Fits Better?

Empirical finding: Many spatial interactions fit power laws better than exponentials over most distance ranges.

Why? Power laws lack a characteristic scale — interaction decays gradually at all distances, rather than dropping sharply beyond a threshold.

Summary

  • The gravity model predicts spatial interaction: $I_{ij} = k \frac{P_i P_j}{d_{ij}^\gamma}$
  • Distance decay exponent $\gamma$ controls sensitivity to distance ($\gamma \approx 1$–$3$)
  • Larger $\gamma$ → stronger distance decay (local interactions dominate)
  • Smaller $\gamma$ → weaker distance decay (globalized interactions)
  • The model applies to migration, trade, commuting, phone calls, retail choice
  • Real flows require calibration and may need corrections for borders, intervening opportunities
  • Power laws (gravity) vs. exponentials (attenuation) — different tail behaviors

References

Anderson, James E., and Eric van Wincoop. 2003. “Gravity with Gravitas: A Solution to the Border Puzzle.” American Economic Review 93 (1): 170–192. https://doi.org/10.1257/000282803321455214

Fotheringham, A. Stewart. 1983. “A New Set of Spatial-Interaction Models: The Theory of Competing Destinations.” Environment and Planning A 15 (1): 15–36. https://doi.org/10.1177/0308518X8301500103

Haynes, Kingsley E., and A. Stewart Fotheringham. 1984. Gravity and Spatial Interaction Models. Beverly Hills: Sage Publications. (No public URL available.)

Huff, David L. 1964. “Defining and Estimating a Trading Area.” Journal of Marketing 28 (3): 34–38. https://doi.org/10.1177/002224296402800307

Krugman, Paul. 1991. “Increasing Returns and Economic Geography.” Journal of Political Economy 99 (3): 483–499. https://doi.org/10.1086/261763

Ravenstein, Ernst Georg. 1885. “The Laws of Migration.” Journal of the Statistical Society of London 48 (2): 167–235. https://doi.org/10.2307/2979181

Reilly, William J. 1931. The Law of Retail Gravitation. New York: W. J. Reilly. (No public URL available; cited in subsequent literature.)

Tinbergen, Jan. 1962. “An Analysis of World Trade Flows.” In Shaping the World Economy: Suggestions for an International Economic Policy, edited by Jan Tinbergen, 262–293. New York: Twentieth Century Fund. (No public URL available.)

Wilson, Alan G. 1971. “A Family of Spatial Interaction Models, and Associated Developments.” Environment and Planning A 3 (1): 1–32. https://doi.org/10.1068/a030001

References