WaywardHouse
The Gravity Model of Trade
The Canada-US bilateral trade relationship is the largest in the world, worth roughly CAD$1.4 trillion in goods in 2023. But that aggregate number conceals enormous provincial variation. Ontario trades nearly twice what Alberta does. The gravity model of international trade — one of the most empirically robust relationships in all of economics — explains why, and quantifies the role of distance, economic mass, and borders in shaping who trades with whom.
Prerequisites: Gravity equation, log linear OLS regression, distance decay, border effect
1. The Question
Why does Canada trade so much with the United States and so much less with equally large economies elsewhere? Why does Ontario account for the largest share of Canada-US trade when Alberta produces more resource wealth per capita? And why, even after three decades of liberalised trade under first the Canada-US Free Trade Agreement and then NAFTA and CUSMA, do provinces still trade many times more with each other than with equivalent US states?
These are not rhetorical questions. They have quantitative answers, derived from one of the most durable empirical regularities in economics: the gravity model of trade. The model is named for its structural resemblance to Newton’s law of universal gravitation. Just as the gravitational force between two objects increases with their masses and decreases with the square of the distance between them, trade between two economies increases with their sizes and decreases with the distance between them. The mathematics is simple. The empirical performance is remarkable. And the residuals — the deviations from the model’s predictions — are among the most informative quantities in economic geography.
This essay derives the gravity equation, fits it to Canada-US provincial trade data, and examines three specific puzzles: the magnitude of the border effect, Alberta’s position in the gravity framework, and what the 2025 US tariff regime implies for bilateral flows when viewed through a model that has been calibrated over decades.
2. The Gravity Equation
The basic model
The gravity model of bilateral trade in its simplest form states that trade between two countries (or regions) $i$ and $j$ is proportional to the product of their economic masses and inversely proportional to the distance between them:
\[T_{ij} = G \cdot \frac{M_i^\alpha \cdot M_j^\beta}{d_{ij}^\gamma}\]where $T_{ij}$ is the value of bilateral trade, $M_i$ and $M_j$ are the economic masses of the trading partners (typically measured by GDP), $d_{ij}$ is the distance between them, and $G$ is a constant capturing all other factors held fixed. The exponents $\alpha$, $\beta$, and $\gamma$ are estimated from data.
The model looks like physics, and it borrows physics’ notation deliberately — but it was not derived from first principles. It emerged empirically. Jan Tinbergen proposed it in 1962 as a description of observed trade patterns, and it has been repeatedly validated across datasets, time periods, and levels of geographic aggregation since. The typical finding is $\alpha \approx \beta \approx 0.9$ (slightly less than proportional to GDP) and $\gamma \approx 1.1$ (distance reduces trade somewhat more than proportionally). More recent studies using modern gravity frameworks and structural estimation tend to find slightly higher distance elasticities, in the range of 1.0 to 1.5, depending on the commodity composition and time period.
Log-linearisation and OLS estimation
The multiplicative form of the gravity equation can be transformed into an additive (linear) form by taking natural logarithms of both sides:
\[\ln T_{ij} = \ln G + \alpha \ln M_i + \beta \ln M_j - \gamma \ln d_{ij} + \varepsilon_{ij}\]This is now a standard ordinary least squares (OLS) regression equation. The dependent variable is the log of bilateral trade; the explanatory variables are the logs of the economic masses and the log of distance. The error term $\varepsilon_{ij}$ captures all variation not explained by mass and distance.
The log-linear form has a useful interpretation. Each coefficient is an elasticity: $\alpha$ is the percentage increase in trade associated with a 1% increase in exporter GDP; $\gamma$ is the percentage decrease in trade associated with a 1% increase in distance. The regression can be estimated with standard software on any dataset that provides bilateral trade values, partner GDPs, and inter-partner distances.
Augmented gravity: the border dummy
The basic model can be augmented with additional variables. The most important augmentation for the Canada-US case is a border dummy variable $B_{ij}$, which equals 1 if $i$ and $j$ are in the same country and 0 if they are in different countries:
\[\ln T_{ij} = \ln G + \alpha \ln M_i + \beta \ln M_j - \gamma \ln d_{ij} + \delta B_{ij} + \varepsilon_{ij}\]The coefficient $\delta$ on the border dummy captures the border effect: the additional reduction in trade caused by crossing an international border, above and beyond what distance alone would predict. When $\delta > 0$, same-country pairs trade more than the gravity model would predict from mass and distance alone. The border effect is typically expressed as a multiplier: $e^\delta$, the factor by which within-country trade exceeds what the model would predict for equivalent international pairs.
3. Canada-US Trade: The Provincial Picture
Canada and the United States together account for the world’s largest bilateral trade relationship by value. In 2023, Canadian goods exports to the United States reached approximately CAD$630 billion, with imports of roughly CAD$470 billion in goods returning northward, for a two-way goods trade total approaching CAD$1.4 trillion. This is larger than any other bilateral goods trade pair in the world, exceeding even US-China and US-Mexico relationships when measured in value terms.
But the provincial distribution of this trade is anything but uniform, and the unevenness is precisely what the gravity model predicts and explains. Ontario, with the largest provincial GDP and the shortest distance to the US manufacturing heartland, accounts for by far the largest share of Canada-US goods trade. Alberta, with the second-largest GDP but much greater distance from major US consumer and manufacturing markets, accounts for substantially less — and even that understates the divergence, because Alberta’s largest export to the US, crude oil, moves predominantly via pipeline and is classified separately from goods trade in standard trade statistics.
The chart below shows estimated Canada-US goods trade by province for 2023, based on provincial trade statistics and federal trade data disaggregated by origin. These are goods-trade figures; they exclude pipeline crude oil and natural gas from Alberta, which are accounted for in energy trade statistics under different commodity classifications.
Ontario’s dominance reflects both GDP size and geographic proximity to Detroit, Buffalo, and the US manufacturing corridor. Alberta’s figure of CAD$175B substantially understates total US-bound flows — crude oil pipeline exports add another CAD$130B+ that does not appear in goods-trade statistics.
4. The Border Effect
The most famous finding in the empirical gravity literature on Canada-US trade is John McCallum’s 1995 result, published in the American Economic Review under the title “National Borders Matter.” McCallum used data from 1988 — the year the Canada-US Free Trade Agreement came into force — and found that Canadian provinces traded, on average, 22 times more with each other than with equivalent US states at the same distance and with comparable economic sizes.
This is the border effect. Twenty-two-fold. At a time when Canada and the United States had already agreed to eliminate tariffs, when the two economies shared language (mostly), legal traditions, currency convertibility, and cultural familiarity. The border still reduced trade by a factor of twenty-two, relative to equivalent within-country flows.
The border effect operates through multiple channels. Different regulatory regimes mean that products certified for one market must be recertified for the other. Different legal systems create transaction costs in contract enforcement. Currency risk, even when managed, imposes hedging costs. Customs procedures impose delays and documentation costs even when duties are zero. Supply chains, marketing networks, and distribution systems have historically developed separately on each side. Trust, familiarity, and business relationships take time to build across borders. Each of these is a modest friction. In aggregate they are substantial.
Since McCallum’s 1995 finding, subsequent research has tracked a gradual decline in the border effect. The full implementation of NAFTA, deeper business integration, the explosion of internet-facilitated trade relationships, and continued regulatory harmonisation reduced the measured border effect from roughly 22x in 1995 to approximately 8-10x by the early 2020s. This is still a large number — the border still matters enormously, even in one of the world’s most integrated bilateral relationships — but the trajectory was downward.
The 2025 imposition of broad US tariffs on Canadian goods represents the first major policy-induced reversal of this trend in three decades. Tariffs raise the cost of international trade directly, reduce the competitiveness of border-crossing relative to domestic alternatives, and disrupt the just-in-time supply chains that have been built on the assumption of frictionless crossing. The estimated border effect multiplier is expected to rise — the chart below shows the historical trajectory and projects the 2025 reversal.
The 2025 tariff spike reverses three decades of integration. A multiplier of 12 means that, controlling for economic size and distance, provinces still trade twelve times more with each other than with equivalent US states. The green dashed line at y=1 represents a theoretically frictionless border — a level no real border in the world has ever achieved.
5. Alberta in the Gravity Framework
When we apply the gravity model to Canadian provincial trade data and plot each province’s actual US trade against the model’s prediction (derived from GDP and distance), Alberta appears as a notable underperformer. Its actual goods-trade volume with the United States is substantially below what the gravity model would predict given Alberta’s GDP and distance to US markets.
This apparent anomaly has several explanations, each of which illuminates a different aspect of the gravity model’s assumptions.
First and most importantly, the standard goods-trade statistics exclude pipeline flows. Alberta’s crude oil exports to the United States — roughly CAD$130 billion per year — move via pipeline and are classified as energy trade rather than goods trade. Natural gas pipeline exports add another CAD$25 billion or so. When these are included, Alberta’s total economic exchange with the United States is substantially larger and aligns much more closely with gravity model predictions. The apparent underperformance is largely a statistical artefact of commodity classification conventions.
Second, distance still matters even controlling for this. Alberta’s major population and economic centres — Edmonton and Calgary — are approximately 2,000 to 4,000 kilometres from the US manufacturing heartland in Detroit, Chicago, and the US Northeast. Ontario’s major centres are 400 to 800 kilometres from those same markets. The gravity model predicts that this distance difference alone produces a substantial trade volume difference even at identical GDPs.
Third, Alberta’s export structure is concentrated in commodities — energy, agriculture, potash, lumber — that move through specific commodity trade channels rather than through the generalised goods-trade statistics that the gravity model was originally calibrated on. The gravity model performs best when trade is diversified across many commodity classes. For highly specialised commodity exporters, the model captures less of the variation.
Alberta (AB) falls below the gravity prediction line, but this reflects the exclusion of pipeline crude oil and gas from goods-trade statistics. Quebec (QC) also underperforms somewhat, reflecting its export structure and distance effects. Saskatchewan (SK) overperforms relative to its GDP due to high-value agricultural and potash exports to specific US destinations.
6. The Alberta Pipeline Correction
The donut chart below shows the composition of Alberta’s total economic flows to the United States, disaggregated by export channel. The majority, by value, moves through pipeline systems that do not appear in standard goods-trade statistics. This has direct implications for how Alberta appears in gravity model analyses: the province is not an underperformer by economic geography — it is an economy whose primary export mode is systematically excluded from the datasets on which gravity models are typically calibrated.
Pipeline exports (crude oil + natural gas) account for roughly 77% of Alberta’s total exports to the United States by value. These flows move through entirely separate trade statistics and infrastructure channels, and are invisible to gravity model analyses based on goods-trade data alone.
Understanding this disaggregation is essential for interpreting any gravity model result involving Alberta. The province is not trading less with the United States than its economic size would predict; it is trading differently — through commodity pipelines rather than trucks and railways crossing border crossings that appear in trade statistics. The gravity model’s prediction is approximately correct when all trade channels are included; the apparent anomaly disappears when the commodity classification conventions are understood.
This distinction also matters for the 2025 tariff analysis. Pipeline crude oil and natural gas do not face the same tariff regime as goods crossing land borders. The US administration’s 2025 tariff orders specifically targeted goods trade at border crossings; energy imports via pipeline were initially treated separately under a different tariff schedule. Alberta’s apparent exposure to the 2025 tariffs is therefore substantially lower than Ontario’s or Quebec’s when measured as a fraction of total US-bound economic flows — though the economic linkages are complex enough that indirect effects through energy pricing and supply chain disruption affect all provinces.
7. What the Model Tells Us
The gravity model of trade is not a causal model. It does not explain why distance reduces trade or why GDP increases it; it describes that these relationships hold empirically with remarkable stability across a wide range of contexts. Its value is precisely in this stability: when an observation deviates significantly from the gravity prediction, something interesting is happening that warrants investigation.
For Canada, the border effect is the most important deviation. For Alberta, the pipeline classification gap is the most important. For the 2025 tariff episode, the gravity model provides a baseline: we can estimate, from the observed relationship between trade barriers and trade volumes, roughly how much a given increase in bilateral trade costs should reduce trade flows. The coefficient $\gamma$ on distance, which proxies for all trade costs that scale with the friction of crossing space, suggests that a 25% tariff imposed as an ad valorem cost on goods trade is equivalent to a substantial increase in effective distance — enough to predict a measurable reduction in the Canada-US border effect’s denominator and an increase in the overall multiplier.
The gravity model was built on peacetime data from decades of gradual trade liberalisation. It has never been calibrated on a data point as sharp as a 25% tariff imposed suddenly on the world’s largest bilateral trading relationship. What it can tell us, with some confidence, is that the direction of the effect is unambiguous: tariffs reduce trade, the reduction is approximately proportional to the tariff level, and the adjustment works through both volume reductions and price changes. The precise magnitude remains to be seen in the data when 2025 and 2026 provincial trade statistics become available.
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
Global Affairs Canada. 2023. Canada’s State of Trade 2023: Inclusive Trade. Ottawa: Government of Canada. https://www.international.gc.ca/transparency-transparence/state-trade-commerce-international/2023.aspx?lang=eng
McCallum, John. 1995. “National Borders Matter: Canada-U.S. Regional Trade Patterns.” American Economic Review 85 (3): 615–623. https://ideas.repec.org/a/aea/aecrev/v85y1995i3p615-23.html
Statistics Canada. 2024. “Canadian International Merchandise Trade: Annual Review 2023.” The Daily, May 9, 2024. Ottawa: Statistics Canada. https://www150.statcan.gc.ca/n1/daily-quotidien/240509/dq240509a-eng.htm
Statistics Canada. 2024. Canadian International Merchandise Trade by Province and Country, and by Product Sections, Customs-Based, Annual. Table 12-10-0173-01. Ottawa: Statistics Canada. https://www150.statcan.gc.ca/t1/tbl1/en/tv.action?pid=1210017301
Statistics Canada. 2024. Focus on Canada and the United States: Trade. Ottawa: Statistics Canada. https://www.statcan.gc.ca/en/topics-start/canada-united-states/trade
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.)