A Global Compass for the Great Divergence: Emissions Versus Production Centres of Gravity 1820–2008

We construct the world's centres of gravity for human population, GDP and CO emissions by taking the best out of five recognised data sources covering the last two centuries. On the basis of a novel distortion&#8208;free representation of these centres of gravity, we find a radical Western shift of GDP and CO emission centres in the nineteenth century, in sharp contrast with the stability of the demographic centre of gravity. Both GDP and emissions trends are reversed in the first half of the twentieth century, after World War I for CO emissions, after World War II for GDP. Since then, both centres are moving eastward at an accelerating speed. These patterns are perfectly consistent with the lead of Western countries starting the industrial revolution, the gradual replacement of coal by oil and gas as alternative sources of energy and the progressive catch&#8208;up of Asian countries in the recent past.


| Cartesian coordinates of world centres of gravity
Assume the surface of the earth is covered by a regular grid of N cells. Each cell i, i = 1,…, N, is identified by the latitude (ϕ) and longitude (λ) of its lower-left corner. For each cell, there is an estimate of the underlying variable V, that is CO 2 emissions (E) for the world emission centre of gravity, GDP (G) for the world economic centre of gravity or population (P) for the world demographic centre of gravity.
The Cartesian coordinates of each centre of gravity are determined according to the three-step methodology previously introduced by Grether and Mathys (2010). First, the share of each cell in the world total is calculated, that is s iV = V i � ∑ N i=1 V i . Second, the Polar coordinates of each grid cell are converted into their corresponding Cartesian coordinates, denoted by x, y and z. For that purpose, the earth is assumed to be a perfect sphere, a reasonable assumption given the approximations affecting the measurement of the underlying variables. Cartesian coordinates may be expressed in kilometres, or as a fraction of the earth's radius, R (6,371 km). 1 Third, the coordinates of the world centre of gravity are obtained as weighted averages of the Cartesian coordinates of each grid cell, using grid cell shares as weights: The obtained point, P * V = (x V , y V , z V ), where V = E, G, P locates within the sphere. The length of the associated vector, with its origin in the earth's centre, is obtained as: This length can be used as a rough indicator of the concentration of the underlying variable on the earth's surface. An extreme concentration in a single point would lead to a gravity centre right on the earth's surface and a length just equal to the earth's radius. 2

| Existing conventions to represent the location of world centres of gravity
The literature on how to map the earth's surface on a two-dimensional plane dates back more than two thousand years (see Snyder, 1987 for a detailed survey including both technical and historical references). There is no universally accepted technique, as every method (cylindrical, conic or azimuthal, and their subcases) presents its shortcomings regarding specific distortions (e.g. on distances, areas or angles). The problem is further compounded here by the fact that the points we are interested in, that is, centres of gravity, are located within the sphere, not on its surface.
To the best of our knowledge, two projection techniques have been proposed for the world centres of gravity, as illustrated by Figure 1. The first, proposed by Grether and Mathys (2010), consists of projecting orthogonally the centre of gravity, P*, upon the earth's surface (Figure 1a). It leaves 1 In a 3-dimensional space where the origin is at the centre of the earth, axis x (projection of the Greenwich meridian) and axis y (projection of the 90°E meridian) define the equatorial plane, and axis z is the north-south polar axis, the corresponding formulas are as follows: x i = Rcos(ϕ i )cos(λ i ), y i = Rcos(ϕ i )sin(λ i ), z i = Rsin(ϕ i ), where R is the earth's radius. See the technical Appendix to Grether and Mathys (2011) for a detailed description. (1) unspecified the technique used to represent the projection point, P 1 , with latitude ϕ 1 . The second technique, proposed by Quah (2011), directly projects the centre of gravity on a cylinder wrapping the globe along the Equator (Figure 1b), which leads to a lower latitude for the projection point, ϕ 2 < ϕ 1 . Both techniques may be criticised on the grounds that they are insensitive to specific directional movements of the centre of gravity, depending on the distribution of the underlying variable over time: Grether and Mathys (2011) do not capture changes of P* along the OP 1 axis; Quah (2011) is insensitive to changes of P* along the QP 2 line. Which type of change matters more in practice is an empirical question, which could guide the choice between these two techniques (or any alternative deemed more relevant depending on the specific variable or time period considered). However, any convention relying on a single two-dimensional map will remain affected by some kind of distortion. That is why we privilege here Cartesian over Geographic coordinates and use two maps instead of one.

| A new, distortion-free convention
The first map, on the left of Figure 2, is consistent with Quah (2011), that is a cylindrical projection. It provides, on the vertical axis, a distortion-free representation of the z Cartesian coordinate described above. The horizontal axis represents longitude, which is subject to distortions, because there is an infinity of (x, y) combinations within the sphere corresponding to the same longitude. The second diagram provides an explicit representation of x and y, with the x(y) axis representing the projection of the Greenwich (90°) meridian. All three Cartesian coordinates are expressed as a fraction of the earth's radius. 3 The combination of these two maps permits describing without distortion any underground movement of the centre of gravity, including those peculiar cases to which previously used conventions are insensitive. Two stylised examples will help to illustrate the complementarity of both maps. In each case, one of the two maps gives a confusing vision of the evolution of the centre of gravity, while the other unveils what actually happens. We dub the first case the 'wipper effect'. It is represented in Figure 3, where the left map suggests that the centre of gravity shifts from A to B, then back again and so forth, as a pendulum covering apparently the same horizontal distance period after period. However, what happens in reality, as shown by the right map, is that the centre of gravity gets ever closer to the centre of the earth, along a zigzag trajectory. Again, this illusion is due to the fact that an infinity of within-sphere (x, y) combinations is compatible with the same longitude.
3 Countries' contours correspond to a Lambert equal-area cylindrical projection in the left map and to an azimuthal projection in the right map. Figures 2-4 limit the number of meridians and parallels to streamline presentation. Consecutive figures with actual results report meridians and parallel every 10°, along with ticks to indicate half of the earth's radius on the -x, y and z axes.
The right map is not exempt from optical illusion either however. In the second case, illustrated in Figure 4, the centre of gravity appears to be going round a regular ellipse on the right map. However, the left map shows that its height above the equatorial plane is regularly decreasing. We call that movement along a downward spiral 'staircase' effect.
Other optical illusions could still be considered but are not reported here for the sake of conciseness, and as we limit the presentation to the two cases which do affect our results. The key point is that, although we keep on using latitudes and longitudes to characterise locations on maps, the centre of gravity is an underground point which is best identified in space by using three Cartesian coordinates rather than two Geographic coordinates.

| DATA SOURCES
Data are obtained by combining five distinct sources. Three databases provide information at the grid level. The HYDE 3.1 database (Klein Goldewijk et al., 2011) provides historical gridded population data from 10,000 B.C to 2005 A.D. Since 1820, the data are available in 10-year intervals and have a

| Population
The only modification of the HYDE database is to extend it from 2005 to 2010. To do so, we apply to each cell's population in 2005 the population growth rate 2005-10 of the corresponding country as obtained from the Maddison database. Country attribution of each cell is obtained by merging HYDE with the Global Database on Administrative Boundaries (GADM, 2012). As explained below, this HYDE gridded population database at a very high degree of resolution provides the basis to extend GDP and emission gridded data backward in time.

| Gross domestic product
First, the G-Econ 2005 gridded GDP data are extended to 2010, using Maddison country real GDP data for growth rates and by relying on the same method for population. Second, we extend the gridded GDP series backward to 1820 by combining the HYDE and Maddison databases by assuming that within-country GDP is uniformly distributed per capita. This allows us to spread national GDP figures from the Maddison database according to the gridded population shares obtained from the HYDE 4 Note that EDGAR covers more carbon dioxide sources, but to correctly match EDGAR with CDIAC (which covers only CO 2 emissions from fossil-fuel consumption and cement production), we retain from EDGAR only CO 2 emissions from IPCC source category 1A (fuel combustion) and 2A (non-metallic mineral processes).
z y x database. The figures obtained are of course an approximation, but given data availability, it is the best way to capture within-country spatial variations backward in time. We then aggregate these 5 arc minutes cells to cells with a 60 arc minutes resolution to match them with the G-Econ data. Finally, we merge the Maddison/HYDE data, covering the decades 1820-2000, with the G-Econ database, which covers the years 1990-2010. 5 Whenever possible, we construct 5-year averages around decimal years to minimise the influence of potential extreme events.

| CO 2 emissions
The procedure is similar to that followed for GDP. First, gridded EDGAR emission data for 2008 are extended to 2012 using 2008-10 and 2010-12 national growth rates obtained from the EDGAR FT2012 database. Second, to extend data backward in time, the HYDE and CDIAC databases are combined assuming emissions per capita are uniformly spread within countries. Then, the data are aggregated to a 60 arc minutes resolution to harmonise with the GDP aggregation level. Finally, we merge the CDIAC/HYDE data, covering the years 1820-1990 with the EDGAR database which covers the years 1970 to 2010. 6 Whenever possible, we construct 5-year averages around decimal years to minimise the influence of potential extreme events. Tables 1 and 2 report summary statistics for each variable. Table 1 reports summary statistics for yearly series (1820 and 2010) that have been aggregated over all cells; Table 2 reports summary statistics at the cell level for 1820 and 2010.

| RESULTS
Figures 5-7 report the two-map diagrams for the three centres of gravity. We remind the reader that the country frontiers are only reported for graphical convenience. Normally, the centre of gravity itself always locates well below the earth's surface. Its height (coordinates along orthogonal meridians) above (within) the equatorial plane is (are) given in the left (right) map. Figure 8a compares the length of the gravity vectors, as the distance between the gravity centre and the earth's centre. It is a rough measure of the concentration of the underlying variable on the earth's surface. It also helps figuring out the radius of the inner-earth imaginary concentric sphere 5 To avoid potential jumps in the final series, we smooth the transition from one database to the other by using a mix of both cell GDP data sets for overlapping decades 1990 and 2000. For the year 1990, we calculate final cell GDP as 70% of Maddison/HYDE cell GDP and 30% of G-Econ cell GDP, while for the year 2000 we calculate it as 30% Maddison/HYDE cell GDP and 70% G-Econ cell GDP. 6 To avoid potential jumps in the final series, we smooth the transition from one database to the other by using a mix of both cell CO 2 data sets for the years 1970, 1980 and 1990, as we did for GDP. For 1970(1980, 1990, we calculate final cell CO 2 emissions as 75% (50%, 25%) of CDIAC/HYDE cell emissions and 25% (50%, 75%) of EDGAR cell emissions.  upon which the centre of gravity locates. Figure 8b compares the speed of the gravity centres, that is the distance they cover per decade. Regarding interpretation of trends, the coordinates of the world centre of gravity being a weighted average of individual cell's coordinates, it is intuitive that changes over time are mostly driven by variations in (large) country shares. 7 To condense presentation, we will only refer to the most important changes in the text below. The interested reader can also refer to the Appendix for the evolution of the share of the largest countries. 7 In theory, within-country variation should also be addressed, but in practice, most of the variation comes from betweencountry changes. See also Grether et al. (2012) for a more in-depth discussion of the underlying drivers and a decomposition of changes in the economic centre of gravity into between-continent and within-continent effects.

| Population
As could be expected, the population centre of gravity is basically located under Asia (Northern India in the left maps and along the Russian-Kazak frontier in the right maps). At the beginning of the period, its length is close to 5,000 km, that is around 0.75R, where R is the earth's radius (6,371 km). This is the result of 0.5R elevation over the equatorial plane (corresponding to a Northern latitude of 30°) and approximately 0.6R rightward orientation on the projection of the 90° meridian (the coordinate along the projection of the Greenwich meridian is almost negligible). In short, human population is initially quite concentrated in the Asian part of the Northern Hemisphere.
The bottom maps reveal a small but steady shift during the sample period, in two distinct phases. During the first phase, which lasts until 1910, the centre of gravity shifts westward, with no latitudinal change. This is consistent with the gradual decline of China and India, whose combined share in world population drops from 55% to 40% along that subperiod. It is also concomitant with a leftward shift of the horizontal component of the left maps and a corresponding decline in the length of the gravity vector by around 15%. That is, human population becomes more homogeneously spread, with a decline in Eastern and a rise in Western locations, in particular the USA.
During the second phase, starting in 1920, there is a clear Southern shift, slightly eastward until 1980, and westward since then. This is consistent with Western countries plateauing in terms of population, the combined share of China and India remaining roughly constant, and a relative increase in Southern countries in East Asia first, and in Africa second. Overall, there is again an increase in the dispersion of human population, although the decline of the length of the gravity vector is more moderate than in the first phase.
These shifts in the demographic gravity centre are consistent with historical trends, but of modest magnitude, with an average speed of less than 200km per decade. The trends exhibited by the other two variables reveal more profound changes.

| Gross domestic product
The trajectory of the economic centre of gravity is also in two phases, but the striking features are that apparent distances covered are far larger than for the demographic centre, whereas the elevation upon the equatorial plane is almost unchanged, with most points locating along the 30°N parallel on left-hand side maps. Starting in 1820, the location is almost identical to the demographic centre of gravity, reflecting the small differences in GDP per capita across countries prior to the industrial revolution. Then, the Big Divergence leads to a strong Western shift of the economic gravity centre, with a speed two to three times faster than for the demographic centre of gravity, and over a longer period. Although the process slows over the 1930s and 1940s, the immediate aftermath of World War II brings its last big Western push, with a 1950 location close to the middle of the Atlantic. During that same subperiod, the combined share of China and India in world GDP dropped from 45% to less than 10%, while that of the USA increased from a few percentage points to more than 25%. Since 1950, the eastward shift has been steady, driven by European reconstruction first and then by the Asian comeback. It seems to accelerate a lot between 2000 and 2010, when the centre of gravity jumps by more than 40° of longitude. However, while interpreting left maps, one has to remember that longitudes are not a precise concept in terms of distances. It does not only depend on latitude (which is here roughly constant), but also on distance from the north-south axis, that is the inward location of the gravity centre within the sphere, which is indicated on the right map. And, between 2000 and 2010, it happens that the centre of gravity gets quite close to the earth centre, ending a continuous decrease in the length of the vector since 1950. As a result, the effective speed in 2010 remains less than in 1950; that is, it is indeed large but not extraordinarily so. This explains the apparent jump and illustrates again how relying on a unique map to represent a three-dimensional movement is misleading.

| CO 2 emissions
The trajectory of the centre of gravity for emissions is even more remarkable than for GDP. It is initially an almost purely British phenomenon, with a centre of gravity locating just underneath the UK, with a length corresponding to 98% of the earth's ratio. As the industrial revolution spreads, and use of coal as the main energy source, this centre begins its descent towards the south-west and the earth's centre. Its most westward location is in 1920, when its projection gets close to the US coast and its length has decreased to 81% of the earth's ratio. During that first period, the speed is similar to that recorded for the economic centre of gravity, although larger for the last two decades of the subperiod (1910 and 1920). Overall, the nineteenth century is a period during which GDP and CO 2 emissions tend to evolve synchronously and westward. This is due to the progressive replacement of the UK by the United States as the major source of world emissions. US dominance peaks in 1920, with a share of 50% of world emissions.
Comparative dynamics of GDP and emissions are altered after World War I. While economic expansion continues its westward trend, the centre of gravity of CO 2 emissions shifts towards the east in 1930 and 1940. This suggests a decoupling between economic activity and pollution, which is probably linked with the early adoption of oil as an alternative, less emission-intensive, source of energy by the United States (i.e. the major polluter), while other major polluters remain more coal-dependent. Indeed, according to Smil (2010), the share of coal in US energy supply peaks in 1910, while it does so only 40 years later in the UK and the USSR. As a result, the share of the United States in world emissions declines strongly in 1930-40, whereas its GDP share remains stable. This explains the earlier reversal of the emission centre of gravity with respect to the economic one. Economic trends remain powerful however, and the US growth spurt following the end of World War II temporarily interrupts the eastern trend in 1950, when both centres of gravity shift westward again, albeit more modestly for the emission centre.
From 1950 onward, the emission centre of gravity is heading east, as is the economic one. This is in line with a decline in US dominance in terms of both GDP and emissions, although the decline is a lot larger for emissions, with a US share in world emissions dropping from above 40% in 1950 to 20% in 1980. This coincides with very large distances covered by the emission centre of gravity, close to | 2829 SAUTER ET Al.
1,000 km per decade, as reported by Figure 8. This suggests again that the transition towards non-coal energy sources such as oil and gas has been quicker in the United States compared to other large emitters (the share of coal falls below 50% as early as 1940 for the United States, but only in 1960 for the UK or Japan, and 1970 for Russia, see Smil, 2010).
During the first two decades following the end of the Cold War, 1990 and 2000, the eastern shift is slowed down, as the US share in world totals either stabilises for emissions or even increases slightly for GDP. This is in line with a pause in the erosion of US dominance and the demise of the USSR. 8 But the movement accelerates again in the last decade, 2010, for both GDP and emissions. This corresponds to the rise of Asian countries, in particular China, which remains heavily dependent on coal as an energy source. By the end of the sample period, the emission centre of gravity locates quite close to the demographic centre of gravity.
In a nutshell, the evolution of the emission centre of gravity suggests radical changes in the spatial distribution of CO 2 emissions on the earth's surface. In two centuries, it shifts from an extremely concentrated location to one which is strikingly similar to the distribution of world population.

AND DISCUSSION
People are unequally spread across the planet's surface. This encapsulates into a location of the demographic centre of gravity which is roughly stable over time, at 0.5R (R = 6,371 km) above the equatorial plane and 0.5R to the right of the Greenwich meridian. If GDP and emissions were equally shared among people, the corresponding centres of gravity would locate at the same place, that is below Northern India, at roughly 70% from the centre of the earth. This is not what has happened during the last two centuries. From there the idea of using distance between the demographic centre of gravity and the comparison one as a proxy for spatial imbalances characterising the per capita distribution of the underlying variable (either GDP or emissions).
More specifically, following Zhao et al. (2003), we define the index of spatial imbalances as the ratio between the actual distance between the demographic centre of gravity and the one it is compared to, and the potential maximum for that distance, that is the length of the demographic centre of gravity vector plus the earth's radius. 9 Applied to GDP and emissions, this leads to the values reported in Figure 9.
What happens for GDP confirms the trend reversal pattern already identified in Figure 6. Spatial imbalances start below 10% and then increase during the Great Divergence, as economic growth takes off in Western countries and their offshoots. The peak is reached in 1950, with an index slightly over 50%. After that, European and then most importantly Asian catch-up decreases spatial imbalances back to 20% at the end of the period.
The temporal pattern for emissions is distinct in that it starts from a large level of close to 50% in 1820. The rest of the trajectory is qualitatively similar to GDP, that is also an inverted-u shape, but with three differences. First, the rising phase is less steep, with a peak at 60%. This is due to the fact that, apart from 8 We warn again the reader against using the left map only to estimate distances covered by the emission centre of gravity in 1990 and 2000. They appear large, in particular in contrast with 1960. However, as shown by the right map, it is a typical 'wiper' effect due to the fact that the centre of gravity locates closer and closer to the earth's centre from 1950 onward. In reality, distances covered are considerably smaller in 1990 or 2000 than in 1960 (see Figure 8). 9 For example, if the demographic centre of gravity is denoted by D, the economic centre of gravity is denoted by G, and the earth's centre is denoted by O, then the index of spatial imbalances for GDP is given by ‖ going west, which increases the index, the centre of gravity is also going southward, which decreases the index. Second, as already noticed in Figure 7, the peak is reached in 1920, not 1950. Third, the decreasing phase is steeper, with a final index of spatial imbalances for emissions around 10% in 2010. Intuitively, if data had been available for earlier centuries, it is probable that the pattern of spatial imbalances for emissions would have looked even more similar to that for GDP. After all, before any country started its industrial revolution, differences in emissions per capita across countries were probably not large, implying a low level of spatial imbalances. This suggests a kind of leading role of emissions with respect to GDP over a long time span.
Although no formal analysis has been performed, the interpretation would be as follows. Start from a preindustrial world where production and emissions are roughly homogeneous across people. Then, technological innovation and the use of fossil fuels give an early boost to Western countries. The impact on emissions is immediate, while the effect on production takes several decades to materialise. During the rest of the nineteenth and early twentieth centuries, as the west industrialises alone, emissions and production go hand in hand. Then, rapid adoption of less emission-intensive energy sources (oil and gas rather than coal) by the United States sends back the emission centre of gravity towards the east as early as the 1930s. Economic activity is characterised by more inertia, but when it starts to shift back as well after 1950, this accelerates further the eastern movement in emissions, also enhanced by the shift of more emission-intensive manufacturing activities towards Asia. As it happens, after a long period of divergence, both the economic and the emission centres of gravity seem to be dragged back to their initial 1820 location determined by demography. 10 The above trends are confirmed when using alternative conventions regarding the smoothing shift from CDIAC to EDGAR data for emissions, or from Maddison to G-Econ data for GDP. Note that temporal patterns for the demographic and economic centres of gravity are similar to those identified by Grether et al. (2012), even though the latter did not rely on the Hyde database to capture withincountry changes in spatial distributions. Moreover, as recently illustrated by Sauter, Grether, and Mathys (2016), alternative hypothesis regarding the spatial distribution of emissions hardly affects overall patterns. Therefore, given data limitations, our results can be considered as reasonably robust. 10 The extreme spatial concentration of emissions at the beginning of the sample period is due to the narrow definition of CDIAC historical data, limited to fossil fuel consumption and cement production only. However, to our knowledge, it is the best historical data on CO 2 emissions available at present.

| CONCLUSIONS
Taking the best of the available databases, this paper proposes for the first time distortion-free representations of the trajectories of the world demographic, economic and emission centres of gravity over the last two centuries. Technological innovation, energy transition, structural change and wars are the main factors underlying observed trends and turning points. In a nutshell, it is as if demography acts like a long run anchor, while emissions and GDP are two outcome variables of a technological diffusion process which increases spatial inequalities during the nineteenth century and progressively decreases them during the twentieth century. The patterns revealed allowing for a quantification of the structural shifts underpinning the Great Divergence, but with a different time-frame depending on the underlying variable. When emissions are used, they suggest a deeper divergence, which starts well before 1820 and leads to an Eastern reversal as early as 1920. These results are in line with current research on the geopolitical origins of capitalism (e.g. Anievas & Nisancioglu, 2015). Based on these results, we suggest the following lines of further research. Trade and FDI activities should be taken into account. For historical reasons first, as they were crucial in promoting the technological innovations at the root of the Industrial Revolution and the Great Divergence (e.g. O'Rourke, Rahman, & Taylor, 2012). For contemporaneous relevance also, as they are key elements of the globalisation process, with a global diffusion of the value chain that has made necessary the computation of trade in value-added flows (see OECD & WTO, 2013). Refining centres of gravity calculations in light of these increased interdependences may unveil interesting trends, for example to trace the potential consequences of the 'belt and road' policy recently adopted by China (The Economist, 2017), or to capture consumption-based rather than production-based CO 2 emission estimates (e.g. Wiebe & Yamano, 2016), a distinction that is at the heart of climate change negotiations today.
Moreover, further research should aim at including even more variables, to capture the many dimensions of human activities and interdependences. To mention just one, in a world that has become more multipolar, large nations rely increasingly on geopolitical power (energy, geography, nuclear and military force) or soft power instruments (trade again, but also diplomacy, advertising, or cultural promotion) to improve their relative positions (e.g. Reynaud & Vauday, 2009or Wang, Cao, & Ge, 2015. This may lead to a frequent rebalancing of socioeconomic influences at the worldwide level. By synthesising the spatial distribution of any variable into a single point, the world centre of gravity approach allows to reveal interesting dynamics within this changing context.