include all equations

docs/ODD/ODD_protocol.md

@@ -115,33 +115,27 @@ Figure 1 Overview of model actions, taken from De Bruijn et al. (2023). The gove
 
 Farmers grow pearl millet, groundnut, sorghum, paddy rice, sugar cane, wheat, cotton, chickpea, maize, green gram, finger millet, sunflower and red gram. Each crop undergoes four growth stages (d1 to d4). The crop coefficient (Kc) is then calculated as follows (Fischer et al., 2021):
 
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 Kc_t =
 \begin{cases} 
   Kc1, & t < d_1 \\
   Kc1 + (t - d_1) \times \frac{Kc2 - Kc1}{d_2}, & d_1 \leq t < d_2 \\
   Kc2, & d_2 \leq t < d_3 \\
   Kc2 + (t - (d_1 + d_2 + d_3)) \times \frac{Kc3 - Kc2}{d_4}, & \text{otherwise}
 \end{cases}
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 where t represents the number of days since planting, and d1 to d4 are the durations of each growth stage. Each crop has their own set of these parameters. At the harvest stage, the actual yield (Ya) is determined based on a maximum reference yield (Yr; Siebert & Döll, 2010), the water-stress reduction factor (KyT), and the ratio of actual evapotranspiration (AET) to potential evapotranspiration (PET) throughout the growth period (Fischer et al., 2021):
 
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 Y_a = Y_r \times \left( 1 - KyT \times \left( 1 - \frac{\sum_{t=0}^{t=h} \text{AET}_t}{\sum_{t=0}^{t=h} \text{PET}_t} \right) \right)
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 After they harvest, yield is converted to income depending on the current market price of that specific crop. At the end of each season, farmers track their yield ratio of that harvest, their potential and actual profits and the 12-month SPEI of that season (from the 12-month SPEI between 1979 and 2016, calibrated from 1981-2010). They also check whether this season’s yield ratio is lower than a moving reference point plus a certain “drought threshold”. The reference point is the 5-year average difference between the reference potential yield and the actual yield, and the additional drought threshold is a calibrated factor. If it is below the moving average reference point and the drought threshold (e.g., 15% below the average yield of the last 5 years), the farmer experiences a drought. In that case, their time since the last drought (table 1) resets and their risk perception rises according to
 
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 \beta_t = c \times 1.6^{-d \times t} + e
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 Where d is  a reduction factor, e is a minimum underestimation of risk and c is the maximum overestimation of risk. The amount that is below the threshold is then multiplied by the yearly average income and added as a two year loan (with interest) to yearly costs as microcredit. 
 
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 Submodel expected utility calculations: 
 
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 \text{SEUT}_{\text{no\_action}} = \int_{p_2}^{p_1} \beta_t \times p_i \times U \left( \sum_{t=0}^{T} \frac{\text{Inc}_{i,x,t}}{(1 + r)^t} \right) dp
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 \text{SEUT}_{\text{tube\_well}} = \int_{p_2}^{p_1} \beta_t \times p_i \times U \left( \sum_{t=0}^{T} \frac{\text{Inc}_{i,x,t}^{\text{adapt}} - C_{t,d}^{\text{adapt}}}{(1 + r)^t} \right) dp
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 \text{SEUT}_{\text{own\_crop\_rotation}} = \int_{p_2}^{p_1} \beta_t \times p_i \times U \left( \sum_{t=0}^{T} \frac{\text{Inc}_{i,x,t} - C_{t,m}^{\text{input}}}{(1 + r)^t} \right) dp
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 \text{EUT}_{\text{own\_crop\_rotation}} = \int_{p_2}^{p_1} p_i \times U \left( \sum_{t=0}^{T} \frac{\text{Inc}_{i,x,t} - C_{t,m}^{\text{input}}}{(1 + r)^t} \right) dp
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 Utility U(x) is a function of expected income Inc and potential adapted income Incadapt per event i and adaptation costs Cadapt. In eq. 2, Cadapt is dependent on groundwater levels and in eq. 4 on current market prices. To calculate the utility of all decisions, we take the integral of the summed and time (t, years) discounted (r) utility under all possible events i with a probability of pi and adjust pi with the subjective risk perception $\beta$t. See table B1 for an overview of all model parameters. The utility U (x) as a function of risk aversion $\sigma$ is as follows: