## 2 Materials and Methods
### Ecosystem Valuation  
The ecosystem valuation (EV) was developed using the opportunity cost methodology based on the main productive activities and soil carbon concentration. The determination of opportunity cost values was based on indicators such as the net benefits (NBs) of each land-cover use and the net present value (NPV) of productive trajectories [50]. In order to obtain the aforementioned indicators, the costs, income, rate of return and net profit of the main land uses in the study area were defined. The land uses considered were crops and pasture. The calculations related to crops were calculated based on the relationship between the yield and the costs of the crop, the net profit was defined by means of the income and the realized investment rate and the rate of return was determined from the profit obtained by the worker. The monetary value of the use of pasture hedges was calculated by means of the dairy sector; most inhabitants have opted for this sector due to the stability of its price. The average values of milk generation per unit of pasture were determined by means of the nutritional parameters of the cattle and the properties of the Kikuyo [51] pasture, which is characteristic of the study site. The yield and profit values of the pastures were calculated using the same relationships as in the cultivated land uses. The cost, income and profit base for crops and pasture was developed based on official information from the Ministry of Agriculture and Livestock (MAG) [51], the National Autonomous Institute of Agricultural Research (INIAP) [52], the Ministry of Environment, Water and Ecological Transition of Ecuador (MAATE) [32] and the Consumer Price Index (CPI) [53]; price variability and projections during the study period were considered. The equations used to develop the aforementioned calculations are detailed below [51]. \[\text{NP}_{i,x}=\text{Q}_{i,x}\cdot\text{P}_{i,x}-\text{CT}_{i,x} \tag{10}\]  
Here, \(\text{NP}_{i,x}\) is the net profit (USD);  
i is the land-use;  
Q is the quantity produced;  
P is the price (USD);  
CT is the total cost (USD). Considering all the characteristic productive trajectories in each of the analysis zones, as well as the net benefits of each of the uses that comprise them, the net present value (NPV) was estimated. \[\text{NPV}_{j,x}=\sum_{t}\frac{\text{NP}_{i,x,t}}{(1+\text{r})^{t}} \tag{11}\]  
Here, NPV is the net present value \(\left(\frac{\text{USD}}{\text{Ra}}\right)\);  
i is the use;  
x is the study area (ha);  
j is the production trajectory;  
\(\left(1+\text{r}\right)^{t}\) is the discount factor for year t; \begin{table}
\begin{tabular}{c c} \hline
**Measure** & **Formula** \\ \hline Kappa index & \(\text{K}=\frac{\text{Pr}_{i,x,t}}{\text{I}-\text{Pr}_{i,x}}\) \\ \hline Profit & \(\text{Gain}:(\text{G}_{\text{ij}})=\text{P}_{j+}-\text{P}_{\text{jj}}\) \\ Degradation & Degradation : \((\text{L}_{\text{ij}})=\text{P}_{j+}-\text{P}_{\text{jj}}\) \\ Persistence & Persistence : \(=\text{P}_{j+}-\text{Gain}\) \\ \hline \end{tabular} D\({}_{\text{ij}}\): correctly classified pixels; C\({}_{\text{i}}\): total classified classes corresponding to a coverage; N: total number of failures; \(\text{Pr}_{i,\text{a}}\): proportion of data matching in both strata; \(\text{P}_{\text{(e)}}\): probability that data agree in both strata; \(\text{P}_{\text{ij}}\): gains at time 2; \(\text{P}_{\text{jj}}\): no change; \(\text{P}_{\text{j}}\): degradation at time 2.  
\end{table} 
Table 1: _Cont._r is the discount rate;  
t is time in years. The formulas described above made it possible to estimate the set of trajectories characteristic of each study area and to establish a comparative analysis of the different opportunity costs over time. The discount rate used to evaluate the costs was 12% [54]; this value is generally used in Ecuador. To relate opportunity costs to carbon content, information from the organic carbon map (GSOC) [34] and the elevation map (DEM) [33] was used.