# Towards a Single Integrative Metric on the Dynamics of Social-Environmental Systems
### Integrative Indicator for SES Dynamics
GDP in 2020 USD is deflated to constant 2011 USD using the consumer price index (CPI) from the Bureau of Labor Statistics. Daily carbon price is available. We used the average carbon price of 9.966 EUR per ton of CO\({}_{2}\) and currency exchange rate of 0.7875 USD per EUR during 2009–2020 due to a lack of price data before 26 October 2009. Livestock was converted to animal unit (AU) equivalents following conversion from livestock to sheep by the FAO ([[http://www.fao.org/3/y4176e/y4176e04.htm](http://www.fao.org/3/y4176e/y4176e04.htm)]([http://www.fao.org/3/y4176e/y4176e04.htm](http://www.fao.org/3/y4176e/y4176e04.htm))). More databases are provided in the tables for potential uses of IMoSES (e.g., livestock, CO\({}_{2}\) emission). All data accessed between June and August 2020.  
where economic production may be any of the widely used indicators for economic development, such as GDP, energy production, agricultural production, etc. ; whereas ecosystem production may be GPP, NPP, grain yield, livestock production, etc. Resource input is the amount of resources that are consumed for an SES to produce and function, such as water or energy input. This generic model needs to consider additional parameters to have different weighing factors when the same resource input is used for both economic and ecosystem production. The parameter (\(\kappa\)) is introduced to reflect the calculation of the two efficiencies:  
\[IMoSES=\frac{\left[Economic\;Production\right]\cdot\left[Ecosystem\;production\right]}{ \left[Resource\;Input\right]^{\kappa}} \tag{2}\]  
where \(\kappa\) is a regulatory parameter and varies from neutral linear (\(\kappa\) = 0), to negative linear (\(\kappa\) = 1), or positive linear (\(\kappa\) < \(-\)1) regulations on the magnitude of IMoSES. Where the unit and weight of economic production and ecosystem production are the same, we would use the sum of these efficiencies. For example, GDP as the most popular indicator of economic systems has a unit of USD yr\({}^{-1}\), whereas NPP as a widely used measure of ecosystem production has a unit of Mg ha\({}^{-1}\) yr\({}^{-1}\). Additionally, the proposed IMoSES needs to consider the land area (A) and population size (POP), so a large country can be compared with a small one, or an SES with high POP can be compared with a less dense one. For example, when GPD is used for economic development, GDP per capita (GDPpc) is recognized to reflect social contributions. Similarly, HANPP (i.e., NPP\(\cdot\)A/POP) is appropriate to represent the ecosystem production of an SES. Using evapotranspiration (ET, mm yr\({}^{-1}\)) as the resource maintaining an SES, IMoSES can be expressed as:  
\[IMoSES=\left[\frac{GDP}{POP}\cdot\frac{(NPP\cdot A)}{POP}\right]\cdot\frac{1}{ ET^{\kappa}} \tag{3}\]  
This calculation can have multiple expressions and several intermediate variables to emphasize different aspects of SES properties and dynamics, including:  
\[=\frac{GDP}{POP}\cdot\frac{1}{POP/A}\cdot\frac{NPP}{ET^{\kappa}} \tag{4}\]  
\[=\frac{GDPpc}{POPd}\cdot WUE\;\;\;\;\text{when}\;\kappa=1 \tag{5}\]  
where A (km\({}^{-2}\)) is the land area of an SES, (NPP\(\cdot\)A, Mg yr\({}^{-1}\)) is the total NPP of the SES, and \(\kappa\) is a sole parameter determining the strength of ET regulations; GDP/POP is known as GDP per capita (GDPpc, USD pers\({}^{-1}\) yr\({}^{-1}\)), POP/A is population density (POPd, pers km\({}^{-2}\)), (NPP\(\cdot\)A)/POP (Mg pers\({}^{-1}\) yr\({}^{-1}\)) is called HANPP [22], and NPP/ET is known as water use efficiency (WUE, mg g\({}^{-1}\)) in ecosystem studies [40; 41]. Using carbon stock information, the unit for NPP can be converted to USD yr\({}^{-1}\) based on monetary valuations of ecosystem services [42], resulting in a unit of USD\({}^{2}\) pers\({}^{-2}\) yr\({}^{-2}\) mm\({}^{-2}\) for IMoSES when \(\kappa\) = 2. To make this unit more meaningful, one can consider the use of the square root of IMoSES (i.e., \(\sqrt{\text{IMoSES}}\)) that will have a unit of USD pers\({}^{-1}\) yr\({}^{-1}\) mm\({}^{-1}\).