MAgPIE - An Open Source land-use modeling framework

4.7.2

created with goxygen 1.4.4

Food demand (15_food)

Description

The food demand module forecasts the food demand vm_dem_food for various agricultural products, which is used in the module 16_demand. Drivers of food demand are population, income, and demographic structure from module 09_drivers, as well as module-specific scenario assumptions. In the case of elastic demand, the model uses the shadow price of agricultural commodities q15_food_demand.m to change food demand and iterates with MAgPIE until a common solution is found. Outputs include next to food demand also projections of anthropometric parameters such as body height and weight distribution, as well as phyiscal activity levels. In addition, this module also provides information about nutrition attributes of foods that can be used in other modules via the interface fm_nutrition_attributes.

Interfaces

Input

Output

Realizations

(A) anthro_iso_jun22

This realization uses per capita income and the demography of the world population (including sex and age classes at the country level, which are provided from module 09_drivers) as drivers. The module estimates food demand on iso-country level taking anthropometric food requirements as well as economic dynamics into account. If flexible demand is activated, the module also uses the shadow prices for agircultural products from the optimization, which are the Lagrange multipliers of the constraint q15_food_demand. The module consists of a standalone food demand model, which is executed before MAgPIE starts. In the case of endogenous demand, the module is iterated with MAgPIE. Besides providing the fooduse of agricultural products, the model also provides a number of output indicators, including the BMI distribution, body weight and height of the population by age and sex, food intake by age group and sex, food waste, dietary composition between livestock products, empty calories (sugar, oil and alcohol), fruits vegetables and nuts, as well as staple calories. The food demand model can be run in standalone mode by running the the file standalone/demand_model.gms. The model is described in (Bodirsky et al. 2020).

The model also includes a number of switches that allow for exogenous diet scenarios, most importantly the shift to the Planetary Health diet (Willett et al. 2019).

The food demand model consists of the following equations, which are not part of MAgPIE.

model m15_food_demand /
      q15_aim,
      q15_budget,
      q15_regr_bmi_shr,
      q15_bmi_shr_verylow,
      q15_bmi_shr_low,
      q15_bmi_shr_medium,
      q15_bmi_shr_medium_high,
      q15_bmi_shr_high,
      q15_bmi_shr_veryhigh,
      q15_bmi_shr_agg,
      q15_intake,
      q15_regr_kcal,
      q15_regr,
      q15_foodtree_kcal_animals,
      q15_foodtree_kcal_processed,
      q15_foodtree_kcal_staples,
      q15_foodtree_kcal_vegetables
  /;

In contrast, the equation q15_food_demand is part of MAgPIE, but not of the food demand model.

\[\begin{multline*} \left(vm\_dem\_food(i2,kfo) + \sum_{ct} f15\_household\_balanceflow(ct,i2,kfo,"dm")\right) \cdot \sum_{ct}\left(fm\_nutrition\_attributes(ct,kfo,"kcal") \cdot 10^{6}\right) \geq \sum_{ct}\left(im\_pop(ct,i2) \cdot p15\_kcal\_pc\_calibrated(ct,i2,kfo)\right) \cdot 365 \end{multline*}\]

Above constraint transforms the fooduse of agricultural products into per capita food demand. vm_dem_food is the fooduse of agricultural products. It is measured in tons of dry matter prior to processing. The multiplication with the nutrition attributes provides the equivalent in calories. While nutrition attributes are assumed to be globally the same (assumption of homogeneous products), a regional balance flow is used to account for current differences in food processing, where some regions get different calories from the same fooduse quantitiy. Depending on the input data, the balance flow may fade out in the future, which implies actual homogeneous products. The subsequent equations belong to the standalone food demand model, which is executed before MAgPIE or iterated with MAgPIE. They are excluded from the run of MAgPIE itself.

\[\begin{multline*} v15\_objective = \sum_{iso}\left( - 10^{6} \cdot v15\_income\_balance(iso)\right) \end{multline*}\]

In principle, the food demand model has only one solution that satifies all equations. Technically, the objective could therefore be chosen arbirtrarily, for the solver to find the single solution. However, if the model is executed outside its domain (e.g. with extreme price shocks), it can happen that real income takes a negative value (because the increase in food value exceeds the available income). To avoid this case, a punishment term v15_income_balance is introduced. It increases the real income, but affects the maximized objective variable negatively, disincentivizing its use in cases where it is not needed.

\[\begin{multline*} v15\_income\_pc\_real\_ppp\_iso(iso) = \sum_{kfo}\left( v15\_kcal\_regr(iso,kfo) \cdot 365 \cdot \left(i15\_prices\_initial\_kcal(iso,kfo)-\sum_{ct,curr\_iter15} p15\_prices\_kcal(ct,iso,kfo,curr\_iter15)\right)\right) + \sum_{ct}im\_gdp\_pc\_ppp\_iso(ct,iso) + v15\_income\_balance(iso) \end{multline*}\]

The budget constraint calculates the real income after a possible price shock. The basic assumption is that increasing prices reduce real income, while decreasing prices increase real income. Through this income effect, higher prices reduce the food demand. The income before the food price shock is im_gdp_pc_ppp_iso. It is reduced by the change in value of the demanded calories under changed prices. In the following, the real income is used to determine food intake, food demand as well as dietary composition. The BMI distribution within the population is calculated using regressions in a hierachical tree to estimate the share of the population falling into a certain BMI class. First, the regression shares are calculated:

\[\begin{multline*} v15\_regr\_overgroups(iso,sex,agegroup15,bmi\_tree15) = i15\_bmi\_intercept(sex,agegroup15,bmi\_tree15) +\frac{ \left(i15\_bmi\_saturation(sex,agegroup15,bmi\_tree15) \cdot v15\_income\_pc\_real\_ppp\_iso(iso)\right) }{ \left(i15\_bmi\_halfsat(sex,agegroup15,bmi\_tree15) + v15\_income\_pc\_real\_ppp\_iso(iso)\right)} \end{multline*}\]

Then, these regression shares are applied to parameterize the hierarchical tree structure:

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"verylow") = v15\_regr\_overgroups(iso,sex,agegroup15,"low") \cdot v15\_regr\_overgroups(iso,sex,agegroup15,"lowsplit") \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"low") = v15\_regr\_overgroups(iso,sex,agegroup15,"low") \cdot \left(1- v15\_regr\_overgroups(iso,sex,agegroup15,"lowsplit")\right) \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"medium") = \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"low") -v15\_regr\_overgroups(iso,sex,agegroup15,"high")\right) \cdot \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"mediumsplit")\right) \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"mediumhigh") = \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"low") -v15\_regr\_overgroups(iso,sex,agegroup15,"high")\right) \cdot v15\_regr\_overgroups(iso,sex,agegroup15,"mediumsplit") \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"high") = v15\_regr\_overgroups(iso,sex,agegroup15,"high") \cdot \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"highsplit")\right) \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"veryhigh") = v15\_regr\_overgroups(iso,sex,agegroup15,"high") \cdot v15\_regr\_overgroups(iso,sex,agegroup15,"highsplit") \end{multline*}\]

From BMI shares of the larger groups (overgroups), we disaggregate to age-specific subgroups.

\[\begin{multline*} v15\_bmi\_shr\_regr(iso,sex,age,bmi\_group15) = \sum_{agegroup2age(agegroup15,age)}\left( v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,bmi\_group15) \right) \end{multline*}\]

Food intake is estimated based on BMI distribution, typical intakes for BMI groups, demographic structure and extra energy requirements for pregnancy and lactation (estimated based on the new-born population size).

\[\begin{multline*} v15\_kcal\_intake\_total\_regr(iso) \cdot \sum_{sex,age,ct} im\_demography(ct,iso,sex,age) = \sum_{ct, sex, age, bmi\_group15}\left( v15\_bmi\_shr\_regr(iso,sex,age,bmi\_group15) \cdot im\_demography(ct,iso,sex,age) \cdot p15\_intake(ct,iso,sex,age,bmi\_group15) \right) + \sum_{ct}i15\_kcal\_pregnancy(ct,iso) \end{multline*}\]

Food demand is based on food intake and a regression based on income, which estimates how much the actual demand is relative to the required intake. The difference between demand and intake is food waste (not explicitly mentioned in this equation)

\[\begin{multline*} v15\_kcal\_regr\_total(iso) = v15\_demand\_regr\left(iso, "overconsumption"\right) \cdot v15\_kcal\_intake\_total\_regr(iso) \end{multline*}\]

This equation estimates key dietary composition regressision factors, such as the share of animal products, empty calories, or fruits, vegetables and nuts.

\[\begin{multline*} v15\_demand\_regr\left(iso, regr15\right) = i15\_dem\_intercept(iso,regr15) +\frac{ \left(i15\_dem\_saturation(iso,regr15) \cdot v15\_income\_pc\_real\_ppp\_iso(iso)\right) }{ \left(i15\_dem\_halfsat(iso,regr15) + v15\_income\_pc\_real\_ppp\_iso(iso)^{i15\_dem\_nonsat(iso,regr15)}\right)} \end{multline*}\]

In the subsequent equations, those parameters are used to determine the dietary composition using a hierachical tree: Total calories are first divided into animal- and plant-based. The plant-based calories are further divided into processed empty calories and nutritious calories, and so on.

\[\begin{multline*} v15\_kcal\_regr(iso,kfo\_ap) = v15\_kcal\_regr\_total(iso) \cdot v15\_demand\_regr\left(iso, "livestockshare"\right) \cdot \sum_{ct}i15\_livestock\_kcal\_structure\_iso(ct,iso,kfo\_ap) \end{multline*}\]

\[\begin{multline*} v15\_kcal\_regr(iso,kfo\_pf) = v15\_kcal\_regr\_total(iso) \cdot \left(1 - v15\_demand\_regr\left(iso, "livestockshare"\right)\right) \cdot v15\_demand\_regr\left(iso, "processedshare"\right) \cdot \sum_{ct}i15\_processed\_kcal\_structure\_iso(ct,iso,kfo\_pf) \end{multline*}\]

\[\begin{multline*} v15\_kcal\_regr(iso,"others") = v15\_kcal\_regr\_total(iso) \cdot \left(1 - v15\_demand\_regr\left(iso, "livestockshare"\right)\right) \cdot \left(1 - v15\_demand\_regr\left(iso, "processedshare"\right)\right) \cdot v15\_demand\_regr\left(iso, "vegfruitshare"\right) \end{multline*}\]

\[\begin{multline*} v15\_kcal\_regr(iso,kfo\_st) = v15\_kcal\_regr\_total(iso) \cdot \left(1 - v15\_demand\_regr\left(iso, "livestockshare"\right)\right) \cdot \left(1 - v15\_demand\_regr\left(iso, "processedshare"\right)\right) \cdot \left(1 - v15\_demand\_regr\left(iso, "vegfruitshare"\right)\right) \cdot \sum_{ct}i15\_staples\_kcal\_structure\_iso(ct,iso,kfo\_st) \end{multline*}\]

Within the major food groups determined by the regressions (animal calories, empty calories, fruits, vegetable and nut calories as well as staple calories), the relative share of individual products (e.g. eggs within animal calories) is assumed to be constant. An exception is the demand for ruminant meat, which declines in the course of the century at the cost of chicken meat.

The calculations are executed in the following order:

In the beginning of each time step, the body height is estimated based on the diets of the previous 15 years. In case that the time step length exceeds 5 years, the consumption is extrapolated using the last two time steps. Body height is estimated using the consumption of proteins and fats, in our case the foodgroups animal products, pulses and oils. To estimate the body size of underaged (0-14 years), we scale the WHO body height recommendations for underaged with the divergence of the 15-19 year old. The body height estimates are repeated again at the end of the time step to improve the results of the extrapolation for cases where the time step length exceeds 5 years.

p15_bodyweight(t,iso,sex,age,bmi_group15)= f15_bmi(sex,age,bmi_group15) * (p15_bodyheight(t,iso,sex,age,"preliminary")/100)**2;

Physical activity levels (PAL) relative to the basic metabolic rate (BMR) are estimated based on physical inactivity levels, assuming PALs for sedentary and medium-active populations of 1.53 and 1.76 respectively:

p15_physical_activity_level(t,iso,sex,age)=
                            im_physical_inactivity(t,iso,sex,age) * 1.53
                            +(1-im_physical_inactivity(t,iso,sex,age)) * 1.76
                            ;
p15_intake(t,iso,sex,age,bmi_group15)=
                        (f15_schofield(sex,age, "intercept")
                        + f15_schofield(sex,age, "slope")*p15_bodyweight(t,iso,sex,age,bmi_group15))
                        * p15_physical_activity_level(t,iso,sex,age);

Pregnancy and lactation require additional food intake. To account for this, newborns are distributed among reproductive women in a population. This number is then multiplied with the extra energy requirements:

i15_kcal_pregnancy(t,iso)=sum(sex,im_demography(t,iso,sex,"0--4")/5) * ((40/66)*845 + (26/66)*675);

Before MAgPIE is executed, the food demand model is executed, at first without price shocks.

Per capita food demand and BMI shares are calibrated so that historical data are met. For this purpose, the residual between the regression fit and the observation is calculated for the historical period. When the historical period ends, the calibration factor is fixed at the value of the last historical time step. Additionally, a second calibration is required to meet the world totals of FAOSTAT food demand for different foods. While the food demand model estimates the demand for all countries of the world, FAOSTAT only covers a subset of countries. To match FAOSTAT totals, the food demand of countries not included in FAOSTAT is calibrated to zero. As this calibration is done ex-post, food demand estimates can still be used for all countries, but MAgPIE only considers demand from FAOSTAT countries.

if (sum(sameas(t_past,t),1) = 1,
    p15_kcal_calib(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))=0) = 0;
    p15_balanceflow_kcal_iso(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))>0) = 0;
    p15_kcal_calib(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))>0) = f15_kcal_pc_iso(t,iso,kfo) - v15_kcal_regr.l(iso, kfo);
    p15_balanceflow_kcal_iso(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))=0) = f15_kcal_pc_iso(t,iso,kfo) - v15_kcal_regr.l(iso, kfo);
    p15_kcal_calib_lastcalibyear(iso,kfo) = p15_kcal_calib(t,iso,kfo);
    p15_balanceflow_kcal_lastcalibyear(iso,kfo) = p15_balanceflow_kcal_iso(t,iso,kfo);
    i15_bmi_shr_calib(t,iso,sex,age,bmi_group15) =
                      f15_bmi_shr_past(t,iso,age,sex,bmi_group15) -
                      v15_bmi_shr_regr.l(iso,sex,age,bmi_group15);
    i15_bmi_shr_calib_lastcalibyear(iso,sex,age,bmi_group15)=
                      i15_bmi_shr_calib(t,iso,sex,age,bmi_group15);
else

Depending on the scenario switch c15_calibscen, the divergence of the demand from the historical data is kept constant or faded out.

    p15_kcal_calib(t,iso,kfo) = p15_kcal_calib_lastcalibyear(iso,kfo) * f15_kcal_calib_fadeout(t,"%c15_calibscen%");

The divergence of the kcal of countries with no FAOSTAT data is kept constant over time.

    p15_balanceflow_kcal_iso(t,iso,kfo) = p15_balanceflow_kcal_lastcalibyear(iso,kfo);

Depending on the scenario switch c15_calibscen, the divergence of the BMI shares from the historical data is kept constant over time or faded out.

   i15_bmi_shr_calib(t,iso,sex,age,bmi_group15) =
                     i15_bmi_shr_calib_lastcalibyear(iso,sex,age,bmi_group15)
                     * f15_kcal_calib_fadeout(t,"%c15_calibscen%");
);
s15_run_diet_postprocessing = 1;
$include "./modules/15_food/anthro_iso_jun22/exodietmacro.gms";
p15_kcal_pc_initial_iso(t,iso,kfo) = p15_kcal_pc_iso(t,iso,kfo);
pm_kcal_pc_initial(t,i,kfo) =  p15_kcal_pc(t,i,kfo);
o15_kcal_regr_initial(t,iso,kfo) = v15_kcal_regr.l(iso,kfo);

Now, MAgPIE is executed.

After one time step of MAgPIE is executed, the shadow prices of the food demand constraint are fed back into the food demand module, and the food demand module is executed once again.

    display "Coupling: Reading out marginal costs from MAgPIE";
    p15_prices_kcal(t,iso,kfo,curr_iter15)=sum(i_to_iso(i,iso), q15_food_demand.m(i,kfo));

  display "starting m15_food_demand in iteration number ", p15_iteration_counter;
  solve m15_food_demand USING nlp MAXIMIZING v15_objective;

It is checked whether MAgPIE and the food demand model have reached sufficient convergence. The criterion for this is whether the real income in the food demand model has changed in any region by more than s15_convergences_measure relative to the last iteration due to changes in food prices from MAgPIE. Moreover, the model aborts when the number of iterations reaches s15_maxiter. As long as the iteration continues, the food prices are transferred from MAgPIE to the food demand model, and the food demand is transferred from the food demand model to MAgPIE.

After each execution of the food demand model, the body height distribution of the population is estimated. The starting point is the body height distribution of the last timestep. The body height estimates of the old period are moved into the subsequent age class (e.g. the 20-24 year old are now 25-29 years old). The age class of 15-19 year old is estimated newly using the body height regressions and the food consumption of the last 15 years.

   p15_bodyheight(t,iso,sex,"15--19","final") =
                     f15_bodyheight_regr_paras(sex,"slope")*
                     (sum(underaged15,
                       p15_kcal_growth_food(t,iso,underaged15)
                     )/3)**f15_bodyheight_regr_paras(sex,"exponent")
                     ;

The bodyheight of the underaged age class (0-14) is assumed to diverge from ‘normal’ body height by the same proportion as the age class of the 15-19 year old.

p15_bodyheight(t,iso,"M","0--4","final")=p15_bodyheight(t,iso,"M","15--19","final")/176*92;
p15_bodyheight(t,iso,"M","5--9","final")=p15_bodyheight(t,iso,"M","15--19","final")/176*125;
p15_bodyheight(t,iso,"M","10--14","final")=p15_bodyheight(t,iso,"M","15--19","final")/176*152;
p15_bodyheight(t,iso,"F","0--4","final")=p15_bodyheight(t,iso,"M","15--19","final")/163*91;
p15_bodyheight(t,iso,"F","5--9","final")=p15_bodyheight(t,iso,"M","15--19","final")/163*124;
p15_bodyheight(t,iso,"F","10--14","final")=p15_bodyheight(t,iso,"M","15--19","final")/163*154;

Finally, the regression outcome is calibrated by a country-specific additive term, which is the residual of the regression fit and observation of the last historical time step.

if (sum(sameas(t_past,t),1) = 1,
  p15_bodyheight_calib(t,iso,sex,age_new_estimated15) = f15_bodyheight(t,iso,sex,age_new_estimated15) - p15_bodyheight(t,iso,sex,age_new_estimated15,"final");
  p15_bodyheight(t,iso,sex,age_new_estimated15,"final") = f15_bodyheight(t,iso,sex,age_new_estimated15);
else
  p15_bodyheight_calib(t,iso,sex,age_new_estimated15)=p15_bodyheight_calib(t-1,iso,sex,age_new_estimated15);
  p15_bodyheight(t,iso,sex,age_new_estimated15,"final")=p15_bodyheight(t,iso,sex,age_new_estimated15,"final")+p15_bodyheight_calib(t,iso,sex,age_new_estimated15)*s15_calibrate;
);

as include statements are not allowed within the if statements, we execute the if statement before, put the result in s15_run_diet_postprocessing, and then execute the if statement again within the included file: This macro is executed twice, once after the presolve and once in the intersolve. It calibrates the values, includes exogenous diet modifications in the postprocessing and aggregates to regional level. The calibration factor is added to the regression value. Negative values that can possibly occur due to calibration are set to zero.

Transition to exogenous Planetary Health diet (PHD) scenarios (Willett et al. 2019): It is possible to define exogenous diet scenarios that replace the regression-based calculation of food intake and demand scenario-dependent targets following the exogenous PHDs, India-specific recommendations or model-internal intake estimates that hit the PHD targets by setting the switch s15_exo_diet to 1, 2 or 3.

if (s15_exo_diet > 0,

1.) In a first step, the exogenous scenario diets are defined by selecting a scenario target for total daily per capita food intake

$ifthen "%c15_kcal_scen%" == "healthy_BMI"
  p15_bmi_shr_target(t,iso,sex,age,bmi_group15)=0;
  p15_bmi_shr_target(t,iso,sex,age,"medium")=1;
  i15_intake_scen_target(t,iso)$(sum((sex,age), im_demography(t,iso,sex,age)) >0 ) =
     (sum((sex, age, bmi_group15), p15_bmi_shr_target(t,iso,sex,age,bmi_group15)*
     im_demography(t,iso,sex,age)*p15_intake(t,iso,sex,age,bmi_group15) )
     + i15_kcal_pregnancy(t,iso))
     / sum((sex,age), im_demography(t,iso,sex,age));
$elseif "%c15_kcal_scen%" == "no_underweight"
  p15_bmi_shr_target(t,iso,sex,age,bmi_group15)=p15_bmi_shr_calibrated(t,iso,sex,age,bmi_group15);
  p15_bmi_shr_target(t,iso,sex,age,"medium")=
            p15_bmi_shr_calibrated(t,iso,sex,age,"verylow")
            + p15_bmi_shr_calibrated(t,iso,sex,age,"low")
            + p15_bmi_shr_calibrated(t,iso,sex,age,"medium");
  p15_bmi_shr_target(t,iso,sex,age,"verylow")=0;
  p15_bmi_shr_target(t,iso,sex,age,"low")=0;
  i15_intake_scen_target(t,iso)$(sum((sex,age), im_demography(t,iso,sex,age)) >0 ) =
     (sum((sex, age, bmi_group15), p15_bmi_shr_target(t,iso,sex,age,bmi_group15)*
     im_demography(t,iso,sex,age)*p15_intake(t,iso,sex,age,bmi_group15) )
     + i15_kcal_pregnancy(t,iso))
     / sum((sex,age), im_demography(t,iso,sex,age));
$elseif "%c15_kcal_scen%" == "no_overweight"
  p15_bmi_shr_target(t,iso,sex,age,bmi_group15)=p15_bmi_shr_calibrated(t,iso,sex,age,bmi_group15);
  p15_bmi_shr_target(t,iso,sex,age,"medium")=
            p15_bmi_shr_calibrated(t,iso,sex,age,"mediumhigh")
            + p15_bmi_shr_calibrated(t,iso,sex,age,"high")
            + p15_bmi_shr_calibrated(t,iso,sex,age,"veryhigh")
            + p15_bmi_shr_calibrated(t,iso,sex,age,"medium");
  p15_bmi_shr_target(t,iso,sex,age,"mediumhigh")=0;
  p15_bmi_shr_target(t,iso,sex,age,"high")=0;
  p15_bmi_shr_target(t,iso,sex,age,"veryhigh")=0;
  i15_intake_scen_target(t,iso)$(sum((sex,age), im_demography(t,iso,sex,age)) >0 ) =
     (sum((sex, age, bmi_group15), p15_bmi_shr_target(t,iso,sex,age,bmi_group15)*
     im_demography(t,iso,sex,age)*p15_intake(t,iso,sex,age,bmi_group15) )
     + i15_kcal_pregnancy(t,iso))
     / sum((sex,age), im_demography(t,iso,sex,age));
$elseif "%c15_kcal_scen%" == "half_overweight"
          p15_bmi_shr_target(t,iso,sex,age,bmi_group15)=p15_bmi_shr_calibrated(t,iso,sex,age,bmi_group15);
          p15_bmi_shr_target(t,iso,sex,age,"medium")=
                    p15_bmi_shr_calibrated(t,iso,sex,age,"mediumhigh")/2
                    + p15_bmi_shr_calibrated(t,iso,sex,age,"high")/2
                    + p15_bmi_shr_calibrated(t,iso,sex,age,"veryhigh")/2
                    + p15_bmi_shr_calibrated(t,iso,sex,age,"medium");
          p15_bmi_shr_target(t,iso,sex,age,"mediumhigh")=p15_bmi_shr_calibrated(t,iso,sex,age,"mediumhigh")/2;
          p15_bmi_shr_target(t,iso,sex,age,"high")=p15_bmi_shr_calibrated(t,iso,sex,age,"high")/2;
          p15_bmi_shr_target(t,iso,sex,age,"veryhigh")=p15_bmi_shr_calibrated(t,iso,sex,age,"veryhigh")/2;
          i15_intake_scen_target(t,iso)$(sum((sex,age), im_demography(t,iso,sex,age)) >0 ) =
             (sum((sex, age, bmi_group15), p15_bmi_shr_target(t,iso,sex,age,bmi_group15)*
             im_demography(t,iso,sex,age)*p15_intake(t,iso,sex,age,bmi_group15) )
             + i15_kcal_pregnancy(t,iso))
             / sum((sex,age), im_demography(t,iso,sex,age));
$elseif "%c15_kcal_scen%" == "no_underweight_half_overweight"
     p15_bmi_shr_target(t,iso,sex,age,bmi_group15)=p15_bmi_shr_calibrated(t,iso,sex,age,bmi_group15);
     p15_bmi_shr_target(t,iso,sex,age,"medium") =
              p15_bmi_shr_calibrated(t,iso,sex,age,"verylow")
               + p15_bmi_shr_calibrated(t,iso,sex,age,"low")
               + p15_bmi_shr_calibrated(t,iso,sex,age,"mediumhigh")/2
               + p15_bmi_shr_calibrated(t,iso,sex,age,"high")/2
               + p15_bmi_shr_calibrated(t,iso,sex,age,"veryhigh")/2
               + p15_bmi_shr_calibrated(t,iso,sex,age,"medium");
     p15_bmi_shr_target(t,iso,sex,age,"verylow")=0;
     p15_bmi_shr_target(t,iso,sex,age,"low")=0;
     p15_bmi_shr_target(t,iso,sex,age,"mediumhigh")=p15_bmi_shr_calibrated(t,iso,sex,age,"mediumhigh")/2;
     p15_bmi_shr_target(t,iso,sex,age,"high")=p15_bmi_shr_calibrated(t,iso,sex,age,"high")/2;
     p15_bmi_shr_target(t,iso,sex,age,"veryhigh")=p15_bmi_shr_calibrated(t,iso,sex,age,"veryhigh")/2;
     i15_intake_scen_target(t,iso)$(sum((sex,age), im_demography(t,iso,sex,age)) >0 ) =
        (sum((sex, age, bmi_group15), p15_bmi_shr_target(t,iso,sex,age,bmi_group15)*
        im_demography(t,iso,sex,age)*p15_intake(t,iso,sex,age,bmi_group15) )
        + i15_kcal_pregnancy(t,iso))
        / sum((sex,age), im_demography(t,iso,sex,age));
$elseif "%c15_kcal_scen%" == "endo"
  i15_intake_scen_target(t,iso) = p15_intake_total(t,iso);
  p15_bmi_shr_target(t,iso,sex,age,bmi_group15) = p15_bmi_shr_calibrated(t,iso,sex,age,bmi_group15);
$else
  i15_intake_scen_target(t,iso) = sum(kfo,i15_intake_EATLancet_all(iso,"%c15_kcal_scen%","%c15_EAT_scen%",kfo));
  p15_bmi_shr_target(t,iso,sex,age,bmi_group15) = 0;
  p15_bmi_shr_target(t,iso,sex,age,"medium") = 1;
$endif

The intake target is adjusted to meet the calorie target

  p15_intake_detailed_scen_target(t,iso,kfo)$(p15_intake_total(t,iso) > 0) =
    p15_intake_detail(t,iso,kfo) / p15_intake_total(t,iso) * i15_intake_scen_target(t,iso);
  p15_intake_detailed_scen_target(t,iso,kfo)$(p15_intake_total(t,iso) = 0) = 0;

2.) The second step defines the daily per capita intake of different food commodities.

if ((s15_exo_diet = 1 or s15_exo_diet = 2),

In case of diet scenarios that are parametrized to a food-specific data set published by the EAT-Lancet Commission (s15_exo_diet=1), this calculation step consists of filling up the scenario target for total daily per capita food intake according to the food-specific calorie intake of non-staple crops of this data set based on exogenous food demand projections. In case that the in step 1.) selected total calorie intake is not equal to total intake of the data set, only the calories for staple crops are modified and calories for non-staple food commodities are preserved. The EAT lancet target values are the same for non-staples irrespective of the calorie target Only non-staples differ

    i15_intake_EATLancet(iso,kfo) =
          i15_intake_EATLancet_all(iso,"2100kcal","%c15_EAT_scen%",kfo);

upper bound for monogastric meat and eggs

    if (s15_exo_monogastric=1,
      p15_intake_detailed_scen_target(t,iso,EAT_monogastrics15)$(p15_intake_detailed_scen_target(t,iso,EAT_monogastrics15) > i15_intake_EATLancet(iso,EAT_monogastrics15))
        = i15_intake_EATLancet(iso,EAT_monogastrics15));

upper bound for ruminant products

    if (s15_exo_ruminant=1,
      p15_intake_detailed_scen_target(t,iso,EAT_ruminants15)$(p15_intake_detailed_scen_target(t,iso,EAT_ruminants15) > i15_intake_EATLancet(iso,EAT_ruminants15))
        = i15_intake_EATLancet(iso,EAT_ruminants15));

target value for fish

    if (s15_exo_fish=1,
        p15_intake_detailed_scen_target(t,iso,"fish") = i15_intake_EATLancet(iso,"fish"));

lower bound for fruits, veggies, nuts and seeds

    if (s15_exo_fruitvegnut=1,
      p15_intake_detailed_scen_target(t,iso,EAT_fruitvegnutseed15)$(p15_intake_detailed_scen_target(t,iso,EAT_fruitvegnutseed15) < i15_intake_EATLancet(iso,EAT_fruitvegnutseed15))
      = i15_intake_EATLancet(iso,EAT_fruitvegnutseed15));

lower bound for pulses

    if (s15_exo_pulses=1,
      p15_intake_detailed_scen_target(t,iso,EAT_pulses15_old)$(p15_intake_detailed_scen_target(t,iso,EAT_pulses15_old) < i15_intake_EATLancet(iso,EAT_pulses15_old))
      = i15_intake_EATLancet(iso,EAT_pulses15_old));

upper bound for sugar

    if (s15_exo_sugar=1,
      p15_intake_detailed_scen_target(t,iso,EAT_sugar15)$(p15_intake_detailed_scen_target(t,iso,EAT_sugar15) > i15_intake_EATLancet(iso,EAT_sugar15))
        = i15_intake_EATLancet(iso,EAT_sugar15));

target value for oils

    if (s15_exo_oils=1,
      p15_intake_detailed_scen_target(t,iso,"oils") = i15_intake_EATLancet(iso,"oils"));

target value for brans

    if (s15_exo_brans=1,
      p15_intake_detailed_scen_target(t,iso,"brans") = i15_intake_EATLancet(iso,"brans"));

target value for single cell protein

    if (s15_exo_scp=1,
      p15_intake_detailed_scen_target(t,iso,"scp") = i15_intake_EATLancet(iso,"scp"));

upper bound for alcohol

    if (s15_exo_alcohol=1,
      p15_intake_detailed_scen_target(t,iso,"alcohol")$(p15_intake_detailed_scen_target(t,iso,"alcohol") > s15_alc_scen*i15_intake_scen_target(t,iso))
        = s15_alc_scen*i15_intake_scen_target(t,iso);
    );
    p15_intake_detailed_scen_target(t,iso,EAT_staples_old)$(sum(EAT_staples2_old, i15_intake_EATLancet(iso,EAT_staples2_old)) > 0) =
            (i15_intake_scen_target(t,iso) - sum(EAT_nonstaples2_old, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2_old))) *
            (i15_intake_EATLancet(iso,EAT_staples_old) / sum(EAT_staples2_old, i15_intake_EATLancet(iso,EAT_staples2_old)))
            ;
    p15_intake_detailed_scen_target(t,iso,EAT_staples_old)$(i15_intake_scen_target(t,iso) - sum(EAT_nonstaples2_old, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2_old)) < 0) =
                    0;
    p15_intake_detailed_scen_target(t,iso,EAT_nonstaples_old)$(i15_intake_scen_target(t,iso) - sum(EAT_nonstaples2_old, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2_old)) < 0) =
             i15_intake_scen_target(t,iso) * p15_intake_detailed_scen_target(t,iso,EAT_nonstaples_old) / sum(EAT_nonstaples2_old, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2_old))
                    ;
    if (smin((iso,kfo), p15_intake_detailed_scen_target(t,iso,kfo)) < 0,
       abort "The parameter p15_intake_detailed_scen_target became negative after calorie balancing.";
    );
elseif s15_exo_diet = 3,

In case of a MAgPIE-specific realization of the EAT Lancet diet (s15_exo_diet=3), model-internal diet projections are constrained by recommended ranges for intake (f15_rec_EATLancet) of different food groups to ensure healthy and sustainable diets according to the EAT-Lancet Commission. After all calorie recommendations for non-staple food groups are satisfied, intake of staple crops is modified such that the in step 1.) selected total calorie intake is met.

  p15_intake_detail(t,iso,kfo)$(sum(EATtar_kfo15(EAT_mtargets15_2,kfo),
                                   sum(EATtar_kfo15_2(EAT_mtargets15_2,kfo2), p15_intake_detail(t,iso,kfo2))) = 0) =
                                   p15_intake_detail(t,iso,kfo) + 1e-6;

The single targets can be set manually via switches (e.g., s15_exo_ruminant, s15_exo_fish, etc.). upper bound for monogastric products

    if (s15_exo_monogastric = 1,
      p15_intake_detailed_scen_target(t,iso,"livst_egg")$(p15_intake_detail(t,iso,"livst_egg")
                                                               > f15_rec_EATLancet(iso,"t_livst_egg","max")
                                                            ) =
             f15_rec_EATLancet(iso,"t_livst_egg","max");
      p15_intake_detailed_scen_target(t,iso,"livst_chick")$(p15_intake_detail(t,iso,"livst_chick")
                                                             > f15_rec_EATLancet(iso,"t_livst_chick","max")
                                                          ) =
           f15_rec_EATLancet(iso,"t_livst_chick","max");
      p15_intake_detailed_scen_target(t,iso,"livst_pig")$(sum(EAT_redmeat15_2, p15_intake_detail(t,iso,EAT_redmeat15_2))
                                                              > f15_rec_EATLancet(iso,"t_redmeat","max")
                                                           ) =
            (p15_intake_detail(t,iso,"livst_pig") / sum(EAT_redmeat15_2, p15_intake_detail(t,iso,EAT_redmeat15_2)))
               * f15_rec_EATLancet(iso,"t_redmeat","max");
       );

upper bound for ruminant products

    if (s15_exo_ruminant = 1,
      p15_intake_detailed_scen_target(t,iso,"livst_rum")$(sum(EAT_redmeat15_2, p15_intake_detail(t,iso,EAT_redmeat15_2))
                                                               > f15_rec_EATLancet(iso,"t_redmeat","max")
                                                            ) =
             (p15_intake_detail(t,iso,"livst_rum") / sum(EAT_redmeat15_2, p15_intake_detail(t,iso,EAT_redmeat15_2)))
                * f15_rec_EATLancet(iso,"t_redmeat","max");
      p15_intake_detailed_scen_target(t,iso,"livst_milk")$(p15_intake_detail(t,iso,"livst_milk")
                                                               > f15_rec_EATLancet(iso,"t_livst_milk","max")
                                                            ) =
             f15_rec_EATLancet(iso,"t_livst_milk","max");
    );

lower bound for fish

    if (s15_exo_fish = 1,
         p15_intake_detailed_scen_target(t,iso,"fish")$(p15_intake_detail(t,iso,"fish")
                                                            < f15_rec_EATLancet(iso,"t_fish","min")
                                                        ) =
               f15_rec_EATLancet(iso,"t_fish","min");
    );

In MAgPIE fruits, vegetables and certain nuts are combined in the ‘other’ food category; Starchy fruits (bananas and plantains) are included in the ‘cassav_sp’ category. The f15_fruitveg2others_kcal_ratio gives the country-level historical share (fixed into the future based on last historic year) of fruits and vegetables in these aggregate categories.

    if (sum(sameas(t_past,t),1) = 1,
        i15_fruit_ratio(t,iso,EAT_special) = f15_fruitveg2others_kcal_ratio(t,iso,EAT_special);
    else
        i15_fruit_ratio(t,iso,EAT_special) = i15_fruit_ratio(t-1,iso,EAT_special);
    );

Separation of starchy fruits (bananas and plantains) and roots (cassava, sweet potato, yams) in the cassav_sp food category

        p15_intake_detail_starchyfruit(t,iso) = i15_fruit_ratio(t,iso,"cassav_sp") * p15_intake_detail(t,iso,"cassav_sp");
        p15_intake_detail_roots(t,iso) = (1 - i15_fruit_ratio(t,iso,"cassav_sp")) * p15_intake_detail(t,iso,"cassav_sp") + p15_intake_detail(t,iso,"potato");
        p15_intake_detailed_scen_starchyfruit(t,iso) = p15_intake_detail_starchyfruit(t,iso);
        p15_intake_detailed_scen_roots(t,iso) = p15_intake_detail_roots(t,iso);
        p15_intake_detailed_scen_target(t,iso,"potato") = p15_intake_detail(t,iso,"potato");

lower bound for fruits, veggies, nuts and seeds

    if (s15_exo_fruitvegnut = 1,

Maximum recommendation for starchy fruits:

        p15_intake_detailed_scen_starchyfruit(t,iso)$(p15_intake_detail_starchyfruit(t,iso) > f15_rec_EATLancet(iso,"t_fruitstarch","max"))
           = f15_rec_EATLancet(iso,"t_fruitstarch","max");

Split the ‘others’ category into fruits plus vegetables and nuts

        p15_intake_detail_fruitveg(t,iso) = i15_fruit_ratio(t,iso,"others") * p15_intake_detail(t,iso,"others") + p15_intake_detailed_scen_starchyfruit(t,iso);
        p15_intake_detail_nuts(t,iso) = (1 - i15_fruit_ratio(t,iso,"others")) * p15_intake_detail(t,iso,"others");
        p15_intake_detailed_scen_fruitveg(t,iso) = p15_intake_detail_fruitveg(t,iso);
        p15_intake_detailed_scen_nuts(t,iso) = p15_intake_detail_nuts(t,iso);

Minimum recommendation for fruits and vegetables:

        p15_intake_detailed_scen_fruitveg(t,iso)$(p15_intake_detail_fruitveg(t,iso) < f15_rec_EATLancet(iso,"t_fruitveg","min"))
            = f15_rec_EATLancet(iso,"t_fruitveg","min");

Extract fruits and vegetables that are part of others category

        p15_intake_detailed_scen_fruitveg(t,iso) = p15_intake_detailed_scen_fruitveg(t,iso) - p15_intake_detailed_scen_starchyfruit(t,iso);

Minimum recommendation for nuts & seeds (a) nuts included in “others” are scaled by the same amount as fruits and vegetables because the food group “others” is treated as homogenous food category

        p15_intake_detailed_scen_nuts(t,iso)$(p15_intake_detail_fruitveg(t,iso) - p15_intake_detailed_scen_starchyfruit(t,iso) > 0)
            = p15_intake_detail_nuts(t,iso) * p15_intake_detailed_scen_fruitveg(t,iso) / (p15_intake_detail_fruitveg(t,iso) - p15_intake_detailed_scen_starchyfruit(t,iso))
          ;

The resulting intake of the “others” category is:

        p15_intake_detailed_scen_target(t,iso,"others") = p15_intake_detailed_scen_fruitveg(t,iso) + p15_intake_detailed_scen_nuts(t,iso);

2. remaining nuts (groundnut) and seeds (rapeseed, sunflower) are scaled up or down towards the EAT nuts target considering scaling of nuts in others.

        p15_intake_detailed_scen_target(t,iso,kfo_ns)$(sum(kfo_ns2, p15_intake_detail(t,iso,kfo_ns2)) > 0)
            = p15_intake_detail(t,iso,kfo_ns) / sum(kfo_ns2, p15_intake_detail(t,iso,kfo_ns2))
                * (f15_rec_EATLancet(iso,"t_nutseeds","min") - p15_intake_detailed_scen_nuts(t,iso));
        p15_intake_detailed_scen_target(t,iso,kfo_ns)$(p15_intake_detailed_scen_target(t,iso,kfo_ns) < 0) = 0;
    );

upper bound for roots

    if (s15_exo_roots = 1,

Maximum recommendation for roots:

        p15_intake_detailed_scen_roots(t,iso)$(p15_intake_detail_roots(t,iso) > f15_rec_EATLancet(iso,"t_roots","max")) =
          f15_rec_EATLancet(iso,"t_roots","max");
        p15_intake_detailed_scen_target(t,iso,"potato")$(p15_intake_detail_roots(t,iso) > f15_rec_EATLancet(iso,"t_roots","max")) =
          p15_intake_detailed_scen_roots(t,iso) / p15_intake_detail_roots(t,iso) * p15_intake_detail(t,iso,"potato");
    );
    p15_intake_detailed_scen_target(t,iso,"cassav_sp") =
      p15_intake_detailed_scen_roots(t,iso) -
      p15_intake_detailed_scen_target(t,iso,"potato") +
      p15_intake_detailed_scen_starchyfruit(t,iso);

lower bound for legumes

    if (s15_exo_pulses = 1,
       p15_intake_detailed_scen_target(t,iso,EAT_pulses15)$(sum(EAT_pulses15_2, p15_intake_detail(t,iso,EAT_pulses15_2))
                                                                 < f15_rec_EATLancet(iso,"t_legumes","min")
                                                               ) =
            (p15_intake_detail(t,iso,EAT_pulses15) / sum(EAT_pulses15_2, p15_intake_detail(t,iso,EAT_pulses15_2)))
             * f15_rec_EATLancet(iso,"t_legumes","min");
    );

upper bound for sugar

    if (s15_exo_sugar = 1,
      p15_intake_detailed_scen_target(t,iso,"sugar")$(p15_intake_detail(t,iso,"sugar")
                                                              > f15_rec_EATLancet(iso,"t_sugar","max")
                                                           ) =
            f15_rec_EATLancet(iso,"t_sugar","max");
    );

upper and lower bound for oils

    if (s15_exo_oils = 1,
       p15_intake_detailed_scen_target(t,iso,"oils")$(p15_intake_detail(t,iso,"oils")
                                                       < f15_rec_EATLancet(iso,"t_oils","min")
                                                      ) =
            f15_rec_EATLancet(iso,"t_oils","min");
       p15_intake_detailed_scen_target(t,iso,"oils")$(p15_intake_detail(t,iso,"oils")
                                                        > f15_rec_EATLancet(iso,"t_oils","max")
                                                     ) =
            f15_rec_EATLancet(iso,"t_oils","max");
    );
 p15_intake_detailed_scen_target(t,iso,kfo_norec) = 0;
    if (s15_exo_alcohol = 1,
       p15_intake_detailed_scen_target(t,iso,"alcohol")$(p15_intake_detailed_scen_target(t,iso,"alcohol") > s15_alc_scen * i15_intake_scen_target(t,iso))
        = s15_alc_scen * i15_intake_scen_target(t,iso);
      );

There is no explicit target for brans in the EATLancet recommendations. It is set to 0 when s15_exo_brans is activated.

    if (s15_exo_brans = 1,
       p15_intake_detailed_scen_target(t,iso,"brans") = 0;
       );

There is no explicit target for single cell protein in the EATLancet recommendations. It is therefore set to 0.

    if (s15_exo_scp = 1,
       p15_intake_detailed_scen_target(t,iso,"scp") = 0;
       );

After all calorie recommendations for non-staple food groups are satisfied, intake of staple crops is now modified such that the above-selected total calorie intake is met.

  p15_intake_detailed_scen_target(t,iso,EAT_staples)$(sum(EAT_staples2, p15_intake_detail(t,iso,EAT_staples2)) > 0) =
      (i15_intake_scen_target(t,iso) - sum(EAT_nonstaples2, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2)))
       * (p15_intake_detail(t,iso,EAT_staples) / sum(EAT_staples2, p15_intake_detail(t,iso,EAT_staples2)));
  p15_intake_detailed_scen_target(t,iso,EAT_staples)$(i15_intake_scen_target(t,iso) - sum(EAT_nonstaples2, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2)) < 0) =
         0;
  p15_intake_detailed_scen_target(t,iso,EAT_nonstaples)$(i15_intake_scen_target(t,iso) - sum(EAT_nonstaples2, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2)) < 0) =
         i15_intake_scen_target(t,iso) * p15_intake_detailed_scen_target(t,iso,EAT_nonstaples) /
         sum(EAT_nonstaples2, p15_intake_detailed_scen_target(t,iso,EAT_nonstaples2));
  if (smin((iso,kfo), p15_intake_detailed_scen_target(t,iso,kfo)) < 0,
     display p15_intake_detailed_scen_target;
     abort "The parameter p15_intake_detailed_scen_target became negative after calorie balancing.";
     );
);

3.) In the third step, the regression-based calculation of intake is faded into the exogenous intake scenario according to a predefined speed of convergence (note that fading should start after the historical time slice of the EAT Lancet diet scenarios (y2010) as defined in i15_exo_foodscen_fader(t,iso)):

  p15_intake_detail(t,iso,kfo) = p15_intake_detail(t,iso,kfo) * (1-i15_exo_foodscen_fader(t,iso))
                      + p15_intake_detailed_scen_target(t,iso,kfo) * i15_exo_foodscen_fader(t,iso);
  p15_bmi_shr_calibrated(t,iso,sex,age,bmi_group15) = p15_bmi_shr_calibrated(t,iso,sex,age,bmi_group15) * (1-i15_exo_foodscen_fader(t,iso))
                      + p15_bmi_shr_target(t,iso,sex,age,bmi_group15) * i15_exo_foodscen_fader(t,iso);
);

4.) The fourth step estimates the calorie supply at household level by multiplying daily per capita calorie intake with the demand2intake ratio that was estimated previously. It assures that if commodities with higher food waste ratio are increasingly consumed, food waste increases.

  p15_kcal_pc_iso(t,iso,kfo) = p15_intake_detail(t,iso,kfo)
                                * p15_demand2intake_ratio_detail(t,iso,kfo);

Total waste share and total intake are adapted to new calculations.

  p15_intake_total(t,iso) = sum(kfo, p15_intake_detail(t,iso,kfo));
  p15_waste_pc(t,iso,kfo) = p15_kcal_pc_iso(t,iso,kfo) - p15_intake_detail(t,iso,kfo);
  p15_demand2intake_ratio(t,iso) = 1$(p15_intake_total(t,iso) = 0) +
         (sum(kfo,p15_kcal_pc_iso(t,iso,kfo)) / p15_intake_total(t,iso))$(
           p15_intake_total(t,iso) > 0);
if (s15_exo_waste = 1,

“Downwards convergence” of regional calorie oversupply due to food waste to the waste reduction target, i.e. only for values that are higher than the target:

    p15_demand2intake_ratio(t,iso)$(p15_demand2intake_ratio(t,iso) > s15_waste_scen )
                      = p15_demand2intake_ratio(t,iso) * (1-i15_exo_foodscen_fader(t,iso))
                        + s15_waste_scen * i15_exo_foodscen_fader(t,iso);
);

waste calculation by crop type

    p15_waste_pc(t,iso,kfo)$(sum(kfo2, p15_waste_pc(t,iso,kfo2))<>0) = p15_waste_pc(t,iso,kfo) / sum(kfo2, p15_waste_pc(t,iso,kfo2)) *
                  (p15_intake_total(t,iso)*p15_demand2intake_ratio(t,iso) - p15_intake_total(t,iso));

Waste ratio is applied

    p15_kcal_pc_iso(t,iso,kfo) = p15_intake_detail(t,iso,kfo) + p15_waste_pc(t,iso,kfo);

Demand intake detail

    p15_demand2intake_ratio_detail(t,iso,kfo)=1$(p15_intake_detail(t,iso,kfo) = 0) +
                  (p15_kcal_pc_iso(t,iso,kfo) / p15_intake_detail(t,iso,kfo))$(p15_intake_detail(t,iso,kfo) > 0);

The country-level parameter p15_kcal_pc_iso is aggregated to the regional level. After removing estimates from countries that are not included in FAOSTAT, the resulting parameter p15_kcal_pc_calibrated is provided to constraint q15_food_demand in the MAgPIE model, which defines the demand for food. Results are aggregated to regions

   p15_kcal_pc(t,i,kfo)$(
      sum(i_to_iso(i,iso),
         im_pop_iso(t,iso)
      ) > 0) =
               sum(i_to_iso(i,iso),
                 p15_kcal_pc_iso(t,iso,kfo)
                 * im_pop_iso(t,iso)
               ) / sum(i_to_iso(i,iso),
                   im_pop_iso(t,iso)
               );
   p15_balanceflow_kcal(t,i,kfo)$(
      sum(i_to_iso(i,iso),
         im_pop_iso(t,iso)
      ) > 0) =
               sum(i_to_iso(i,iso),
                 p15_balanceflow_kcal_iso(t,iso,kfo)
                 * im_pop_iso(t,iso)
               ) / sum(i_to_iso(i,iso),
                   im_pop_iso(t,iso)
               );

Finally, countries with zero food demand according to FAOSTAT are calibrated down to zero to match FAO world totals. Values are rounded to avoid path dependencies of MAgPIE solver.

   p15_kcal_pc_calibrated(t,i,kfo) = p15_kcal_pc(t,i,kfo) + p15_balanceflow_kcal(t,i,kfo);
   p15_kcal_pc_calibrated(t,i,kfo) = round(p15_kcal_pc_calibrated(t,i,kfo), 2);
   p15_kcal_pc_calibrated(t,i,kfo)$(p15_kcal_pc_calibrated(t,i,kfo) < 0) = 0;

Limitations There are no known limitations.

(B) anthropometrics_jan18

The realization anthropometrics_jan18 uses per capita income and the demography of the world population (including sex and age classes at the country level, which are provided from module 09_drivers) as drivers. The module estimates food demand on iso-country level taking anthropometric food requirements as well as economic dynamics into account. If flexible demand is activated, the module also uses the shadow prices for agircultural products from the optimization, which are the Lagrange multipliers of the constraint q15_food_demand. The module consists of a standalone food demand model, which is executed before MAgPIE starts. In the case of endogenous demand, the module is iterated with MAgPIE. Besides providing the fooduse of agricultural products, the model also provides a number of output indicators, including the BMI distribution, body weight and height of the population by age and sex, food intake by age group and sex, food waste, dietary composition between livestock products, empty calories (sugar, oil and alcohol), fruits vegetables and nuts, as well as staple calories. The food demand model can be run in standalone mode by running the the file standalone/demand_model.gms. The model is described in (Bodirsky et al. 2020)

The food demand model consists of the following equations, which are not part of MAgPIE.

model m15_food_demand /
      q15_aim,
      q15_budget,
      q15_regr_bmi_shr,
      q15_bmi_shr_verylow,
      q15_bmi_shr_low,
      q15_bmi_shr_medium,
      q15_bmi_shr_medium_high,
      q15_bmi_shr_high,
      q15_bmi_shr_veryhigh,
      q15_bmi_shr_agg,
      q15_intake,
      q15_regr_kcal,
      q15_regr,
      q15_foodtree_kcal_animals,
      q15_foodtree_kcal_processed,
      q15_foodtree_kcal_staples,
      q15_foodtree_kcal_vegetables
  /;

In contrast, the equation q15_food_demand is part of MAgPIE, but not of the food demand model.

\[\begin{multline*} \left(vm\_dem\_food(i2,kfo) + \sum_{ct} f15\_household\_balanceflow(ct,i2,kfo,"dm")\right) \cdot \sum_{ct}\left(fm\_nutrition\_attributes(ct,kfo,"kcal") \cdot 10^{6}\right) \geq \sum_{ct}\left(im\_pop(ct,i2) \cdot p15\_kcal\_pc\_calibrated(ct,i2,kfo)\right) \cdot 365 \end{multline*}\]

Above constraint transforms the fooduse of agricultural products into per capita food demand. vm_dem_food is the fooduse of agricultural products. It is measured in tons of dry matter prior to processing. The multiplication with the nutrition attributes provides the equivalent in calories. While nutrition attributes are assumed to be globally the same (assumption of homogeneous products), a regional balance flow is used to account for current differences in food processing, where some regions get different calories from the same fooduse quantitiy. Depending on the input data, the balance flow may fade out in the future, which implies actual homogeneous products. The subsequent equations belong to the standalone food demand model, which is executed before MAgPIE or iterated with MAgPIE. They are excluded from the run of MAgPIE itself.

\[\begin{multline*} v15\_objective = \sum_{iso}\left( - 10^{6} \cdot v15\_income\_balance(iso)\right) \end{multline*}\]

In principle, the food demand model has only one solution that satifies all equations. Technically, the objective could therefore be chosen arbirtrarily, for the solver to find the single solution. However, if the model is executed outside its domain (e.g. with extreme price shocks), it can happen that real income takes a negative value (because the increase in food value exceeds the available income). To avoid this case, a punishment term v15_income_balance is introduced. It increases the real income, but affects the maximized objective variable negatively, disincentivizing its use in cases where it is not needed.

\[\begin{multline*} v15\_income\_pc\_real\_ppp\_iso(iso) = \sum_{kfo}\left( v15\_kcal\_regr(iso,kfo) \cdot 365 \cdot \left(i15\_prices\_initial\_kcal(iso,kfo)-\sum_{ct,curr\_iter15} p15\_prices\_kcal(ct,iso,kfo,curr\_iter15)\right)\right) + \sum_{ct}im\_gdp\_pc\_ppp\_iso(ct,iso) + v15\_income\_balance(iso) \end{multline*}\]

The budget constraint calculates the real income after a possible price shock. The basic assumption is that increasing prices reduce real income, while decreasing prices increase real income. Through this income effect, higher prices reduce the food demand. The income before the food price shock is im_gdp_pc_ppp_iso. It is reduced by the change in value of the demanded calories under changed prices. In the following, the real income is used to determine food intake, food demand as well as dietary composition. The BMI distribution within the population is calculated using regressions in a hierachical tree to estimate the share of the population falling into a certain BMI class. First, the regression shares are calculated:

\[\begin{multline*} v15\_regr\_overgroups(iso,sex,agegroup15,bmi\_tree15) = i15\_bmi\_intercept(sex,agegroup15,bmi\_tree15) +\frac{ \left(i15\_bmi\_saturation(sex,agegroup15,bmi\_tree15) \cdot v15\_income\_pc\_real\_ppp\_iso(iso)\right) }{ \left(i15\_bmi\_halfsat(sex,agegroup15,bmi\_tree15) + v15\_income\_pc\_real\_ppp\_iso(iso)\right)} \end{multline*}\]

Then, these regression shares are applied to parameterize the hierarchical tree structure:

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"verylow") = v15\_regr\_overgroups(iso,sex,agegroup15,"low") \cdot v15\_regr\_overgroups(iso,sex,agegroup15,"lowsplit") \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"low") = v15\_regr\_overgroups(iso,sex,agegroup15,"low") \cdot \left(1- v15\_regr\_overgroups(iso,sex,agegroup15,"lowsplit")\right) \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"medium") = \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"low") -v15\_regr\_overgroups(iso,sex,agegroup15,"high")\right) \cdot \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"mediumsplit")\right) \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"mediumhigh") = \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"low") -v15\_regr\_overgroups(iso,sex,agegroup15,"high")\right) \cdot v15\_regr\_overgroups(iso,sex,agegroup15,"mediumsplit") \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"high") = v15\_regr\_overgroups(iso,sex,agegroup15,"high") \cdot \left(1-v15\_regr\_overgroups(iso,sex,agegroup15,"highsplit")\right) \end{multline*}\]

\[\begin{multline*} v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,"veryhigh") = v15\_regr\_overgroups(iso,sex,agegroup15,"high") \cdot v15\_regr\_overgroups(iso,sex,agegroup15,"highsplit") \end{multline*}\]

From BMI shares of the larger groups (overgroups), we disaggregate to age-specific subgroups.

\[\begin{multline*} v15\_bmi\_shr\_regr(iso,sex,age,bmi\_group15) = \sum_{agegroup2age(agegroup15,age)}\left( v15\_bmi\_shr\_overgroups(iso,sex,agegroup15,bmi\_group15) \right) \end{multline*}\]

Food intake is estimated based on BMI distribution, typical intakes for BMI groups, demographic structure and extra energy requirements for pregnancy and lactation (estimated based on the new-born population size).

\[\begin{multline*} v15\_kcal\_intake\_total\_regr(iso) \cdot \sum_{sex,age,ct} im\_demography(ct,iso,sex,age) = \sum_{ct, sex, age, bmi\_group15}\left( v15\_bmi\_shr\_regr(iso,sex,age,bmi\_group15) \cdot im\_demography(ct,iso,sex,age) \cdot p15\_intake(ct,iso,sex,age,bmi\_group15) \right) + \sum_{ct}i15\_kcal\_pregnancy(ct,iso) \end{multline*}\]

Food demand is based on food intake and a regression based on income, which estimates how much the actual demand is relative to the required intake. The difference between demand and intake is food waste (not explicitly mentioned in this equation)

\[\begin{multline*} v15\_kcal\_regr\_total(iso) = v15\_demand\_regr\left(iso, "overconsumption"\right) \cdot v15\_kcal\_intake\_total\_regr(iso) \end{multline*}\]

This equation estimates key dietary composition regressision factors, such as the share of animal products, empty calories, or fruits, vegetables and nuts.

\[\begin{multline*} v15\_demand\_regr\left(iso, regr15\right) = i15\_dem\_intercept(iso,regr15) +\frac{ \left(i15\_dem\_saturation(iso,regr15) \cdot v15\_income\_pc\_real\_ppp\_iso(iso)\right) }{ \left(i15\_dem\_halfsat(iso,regr15) + v15\_income\_pc\_real\_ppp\_iso(iso)^{i15\_dem\_nonsat(iso,regr15)}\right)} \end{multline*}\]

In the subsequent equations, those parameters are used to determine the dietary composition using a hierachical tree: Total calories are first divided into animal- and plant-based. The plant-based calories are further divided into processed empty calories and nutritious calories, and so on.

\[\begin{multline*} v15\_kcal\_regr(iso,kfo\_ap) = v15\_kcal\_regr\_total(iso) \cdot v15\_demand\_regr\left(iso, "livestockshare"\right) \cdot \sum_{ct}i15\_livestock\_kcal\_structure\_iso(ct,iso,kfo\_ap) \end{multline*}\]

\[\begin{multline*} v15\_kcal\_regr(iso,kfo\_pf) = v15\_kcal\_regr\_total(iso) \cdot \left(1 - v15\_demand\_regr\left(iso, "livestockshare"\right)\right) \cdot v15\_demand\_regr\left(iso, "processedshare"\right) \cdot \sum_{ct}i15\_processed\_kcal\_structure\_iso(ct,iso,kfo\_pf) \end{multline*}\]

\[\begin{multline*} v15\_kcal\_regr(iso,"others") = v15\_kcal\_regr\_total(iso) \cdot \left(1 - v15\_demand\_regr\left(iso, "livestockshare"\right)\right) \cdot \left(1 - v15\_demand\_regr\left(iso, "processedshare"\right)\right) \cdot v15\_demand\_regr\left(iso, "vegfruitshare"\right) \end{multline*}\]

\[\begin{multline*} v15\_kcal\_regr(iso,kfo\_st) = v15\_kcal\_regr\_total(iso) \cdot \left(1 - v15\_demand\_regr\left(iso, "livestockshare"\right)\right) \cdot \left(1 - v15\_demand\_regr\left(iso, "processedshare"\right)\right) \cdot \left(1 - v15\_demand\_regr\left(iso, "vegfruitshare"\right)\right) \cdot \sum_{ct}i15\_staples\_kcal\_structure\_iso(ct,iso,kfo\_st) \end{multline*}\]

Within the major food groups determined by the regressions (animal calories, empty calories, fruits, vegetable and nut calories as well as staple calories), the relative share of individual products (e.g. eggs within animal calories) is assumed to be constant. An exception is the demand for ruminant meat, which declines in the course of the century at the cost of chicken meat.

The calculations are executed in the following order:

In the beginning of each time step, the body height is estimated based on the diets of the previous 15 years. In case that the time step length exceeds 5 years, the consumption is extrapolated using the last two time steps. Body height is estimated using the consumption of proteins and fats, in our case the foodgroups animal products, pulses and oils. To estimate the body size of underaged (0-14 years), we scale the WHO body height recommendations for underaged with the divergence of the 15-19 year old. The body height estimates are repeated again at the end of the time step to improve the results of the extrapolation for cases where the time step length exceeds 5 years.

p15_bodyweight(t,iso,sex,age,bmi_group15)= f15_bmi(sex,age,bmi_group15) * (p15_bodyheight(t,iso,sex,age,"preliminary")/100)**2;

Physical activity levels (PAL) relative to the basic metabolic rate (BMR) are estimated based on physical inactivity levels, assuming PALs for sedentary and medium-active populations of 1.53 and 1.76 respectively:

p15_physical_activity_level(t,iso,sex,age)=
                            im_physical_inactivity(t,iso,sex,age) * 1.53
                            +(1-im_physical_inactivity(t,iso,sex,age)) * 1.76
                            ;
p15_intake(t,iso,sex,age,bmi_group15)=
                        (f15_schofield(sex,age, "intercept")
                        + f15_schofield(sex,age, "slope")*p15_bodyweight(t,iso,sex,age,bmi_group15))
                        * p15_physical_activity_level(t,iso,sex,age);

Pregnancy and lactation require additional food intake. To account for this, newborns are distributed among reproductive women in a population. This number is then multiplied with the extra energy requirements:

i15_kcal_pregnancy(t,iso)=sum(sex,im_demography(t,iso,sex,"0--4")/5) * ((40/66)*845 + (26/66)*675);

Before MAgPIE is executed, the food demand model is executed, at first without price shocks.

Per capita food demand and BMI shares are calibrated so that historical data are met. For this purpose, the residual between the regression fit and the observation is calculated for the historical period. When the historical period ends, the calibration factor is fixed at the value of the last historical time step. Additionally, a second calibration is required to meet the world totals of FAOSTAT food demand for different foods. While the food demand model estimates the demand for all countries of the world, FAOSTAT only covers a subset of countries. To match FAOSTAT totals, the food demand of countries not included in FAOSTAT is calibrated to zero. As this calibration is done ex-post, food demand estimates can still be used for all countries, but MAgPIE only considers demand from FAOSTAT countries.

if (sum(sameas(t_past,t),1) = 1,
    p15_kcal_calib(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))=0) = 0;
    p15_balanceflow_kcal_iso(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))>0) = 0;
    p15_kcal_calib(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))>0) = f15_kcal_pc_iso(t,iso,kfo) - v15_kcal_regr.l(iso, kfo);
    p15_balanceflow_kcal_iso(t,iso,kfo)$(sum(kfo2,f15_kcal_pc_iso(t,iso,kfo2))=0) = f15_kcal_pc_iso(t,iso,kfo) - v15_kcal_regr.l(iso, kfo);
    p15_kcal_calib_lastcalibyear(iso,kfo) = p15_kcal_calib(t,iso,kfo);
    p15_balanceflow_kcal_lastcalibyear(iso,kfo) = p15_balanceflow_kcal_iso(t,iso,kfo);
    i15_bmi_shr_calib(t,iso,sex,age,bmi_group15) =
                      f15_bmi_shr_past(t,iso,age,sex,bmi_group15) -
                      v15_bmi_shr_regr.l(iso,sex,age,bmi_group15);
    i15_bmi_shr_calib_lastcalibyear(iso,sex,age,bmi_group15)=
                      i15_bmi_shr_calib(t,iso,sex,age,bmi_group15);
else

Depending on the scenario switch c15_calibscen, the divergence of the demand from the historical data is kept constant or faded out.

    p15_kcal_calib(t,iso,kfo) = p15_kcal_calib_lastcalibyear(iso,kfo) * f15_kcal_calib_fadeout(t,"%c15_calibscen%");

The divergence of the kcal of countries with no FAOSTAT data is kept constant over time.

    p15_balanceflow_kcal_iso(t,iso,kfo) = p15_balanceflow_kcal_lastcalibyear(iso,kfo);

Depending on the scenario switch c15_calibscen, the divergence of the BMI shares from the historical data is kept constant over time or faded out.

   i15_bmi_shr_calib(t,iso,sex,age,bmi_group15) =
                     i15_bmi_shr_calib_lastcalibyear(iso,sex,age,bmi_group15)
                     * f15_kcal_calib_fadeout(t,"%c15_calibscen%");
);

The calibration factor is added to the regression value.

   p15_kcal_pc_iso(t,iso,kfo) =
          v15_kcal_regr.l(iso,kfo) + p15_kcal_calib(t,iso,kfo) * s15_calibrate;

Negative values that can possibly occur due to calibration are set to zero.

   p15_kcal_pc_iso(t,iso,kfo)$(p15_kcal_pc_iso(t,iso,kfo)<0) = 0;

The country-level parameter p15_kcal_pc_iso is aggregated to the regional level. After removing estimates from countries that are not included in FAOSTAT, the resulting parameter p15_kcal_pc_calibrated is provided to constraint q15_food_demand in the MAgPIE model, which defines the demand for food.

Transition to exogenous EAT Lancet diet scenarios: It is possible to define exogenous diet scenarios that replace the regression-based calculation of food intake and demand according to a predefined speed of convergence from p15_kcal_pc_calibrated(t,i,kfo) to the scenario-dependent target i15_kcal_pc_scen_target(t,i,kfo) by setting the switch s15_exo_diet to 1.

if(s15_exo_diet = 1,
$ifthen "%c15_exo_foodscen%" == "lin_zero_20_30"
  i15_intake_EATLancet_all(i,kcal_scen15,EAT_scen15,kfo) = f15_intake_EATLancet("y2030",i,kcal_scen15,EAT_scen15,kfo);
$else
  i15_intake_EATLancet_all(i,kcal_scen15,EAT_scen15,kfo) = f15_intake_EATLancet("y2050",i,kcal_scen15,EAT_scen15,kfo);
$endif

1.) In a first step, the exogenous scenario diets are defined by selecting a scenario target for total daily per capita food intake and by choosing food-specific dietary patterns:

$ifthen "%c15_kcal_scen%" == "healthy_BMI"
  i15_intake_scen_target(t,i) = sum(i_to_iso(i,iso),
        sum((sex, age), im_demography(t,iso,sex,age)*p15_intake(t,iso,sex,age,"medium") )
         + i15_kcal_pregnancy(t,iso)
         ) / sum(i_to_iso(i,iso),
             sum((sex,age), im_demography(t,iso,sex,age))
         );
  i15_intake_EATLancet(i,kfo) =
        i15_intake_EATLancet_all(i,"2100kcal","%c15_EAT_scen%",kfo);
$else
  i15_intake_EATLancet(i,kfo) =
        i15_intake_EATLancet_all(i,"%c15_kcal_scen%","%c15_EAT_scen%",kfo);
        i15_intake_scen_target(t,i) = sum(kfo,i15_intake_EATLancet(i,kfo));
$endif

2.) The second step defines the daily per capita intake of different food commodities by filling up the scenario target for total daily per capita food intake according to different scenario assumptions on dietary patterns. Calories for staple crops can be modified in order to meet the total calorie target.

p15_intake_detailed_regr(t,i,kfo) = p15_kcal_pc_calibrated(t,i,kfo)
    /(f15_calib_fsupply(i)*f15_overcons_FAOwaste(i,kfo)*p15_foodwaste_growth(t,i));
p15_intake_detailed_scen_target(t,i,EAT_nonstaples) = i15_intake_EATLancet(i,EAT_nonstaples);
if(s15_alc_scen>0,
p15_intake_detailed_scen_target(t,i,"alcohol") = p15_intake_detailed_regr(t,i,"alcohol");
p15_intake_detailed_scen_target(t,i,"alcohol")$(p15_intake_detailed_scen_target(t,i,"alcohol") > s15_alc_scen*i15_intake_scen_target(t,i))
  = s15_alc_scen*i15_intake_scen_target(t,i);
);
p15_intake_detailed_scen_target(t,i,EAT_staples) = (
          i15_intake_scen_target(t,i) - sum(EAT_nonstaples,i15_intake_EATLancet(i,EAT_nonstaples)) )*(
          i15_intake_EATLancet(i,EAT_staples)/sum(EAT_staples2,i15_intake_EATLancet(i,EAT_staples2)) );

3.) The third step estimates the calorie supply at household level by multiplying daily per capita calorie intake with a ratio of supply to intake (f15_overcons_FAOwaste(i,kfo)), based on FAO estimates on historical food waste at consumption level and food conversion factors, and with a calibration factor f15_calib_fsupply(i). This factor ensures that the estimated food supply (based on MAgPIE calorie intake, FAO waste shares and food converion factors) is calibrated to FAO food supply for the only historical time slice of the EAT Lancet diet scenarios (y2010). A multiplicative factor accounts for increases in food waste over time relative to the only historical time slice of the EAT Lancet diet scenarios, according to the regression-based approach.

i15_kcal_pc_scen_target(t,i,kfo) = (f15_calib_fsupply(i)*f15_overcons_FAOwaste(i,kfo)
                                    *p15_intake_detailed_scen_target(t,i,kfo))
                                    *p15_foodwaste_growth(t,i);

4.) In the last step, the regression-based calculation of daily per capita food demand is faded into the exogenous diet scenario according to a predefined speed of convergence (note that fading should start after the historical time slice of the EAT Lancet diet scenarios (y2010) as defined in i15_exo_foodscen_fader(t,i)):

p15_kcal_pc_calibrated_orig(t,i,kfo) = p15_kcal_pc_calibrated(t,i,kfo);
p15_kcal_pc_calibrated(t,i,kfo) = p15_kcal_pc_calibrated_orig(t,i,kfo) * (1-i15_exo_foodscen_fader(t,i))
                        + i15_kcal_pc_scen_target(t,i,kfo) * i15_exo_foodscen_fader(t,i);
);

Now, MAgPIE is executed.

After one time step of MAgPIE is executed, the shadow prices of the food demand constraint are fed back into the food demand module, and the food demand module is executed once again.

If s15_elastic_demand is 0, MAgPIE is not executed again for this time step. In case that s15_elastic_demand is 1, it is checked whether MAgPIE and the food demand model have reached sufficient convergence. The criterion for this is whether the real income in the food demand model has changed in any region by more than s15_convergences_measure relative to the last iteration due to changes in food prices from MAgPIE. Moreover, the model aborts when the number of iterations reaches s15_maxiter. As long as the iteration continues, the food prices are transferred from MAgPIE to the food demand model, and the food demand is transferred from the food demand model to MAgPIE.

Transition to exogenous EAT Lancet diet scenarios: A part of the definition of exogenous diet scenarios is already accomplished in the presolve.gms, where the parameters i15_intake_scen_target(t,i) and i15_intake_EATLancet(i,kfo) are calculated.

if(s15_exo_diet = 1,
  p15_intake_detailed_regr(t,i,kfo) = p15_kcal_pc_calibrated(t,i,kfo)
     /(f15_calib_fsupply(i)*f15_overcons_FAOwaste(i,kfo)*p15_foodwaste_growth(t,i));
  if(s15_alc_scen>0,
    p15_intake_detailed_scen_target(t,i,"alcohol") = p15_intake_detailed_regr(t,i,"alcohol");
    p15_intake_detailed_scen_target(t,i,"alcohol")$(p15_intake_detailed_scen_target(t,i,"alcohol") > s15_alc_scen*i15_intake_scen_target(t,i))
     = s15_alc_scen*i15_intake_scen_target(t,i);
     );
  p15_intake_detailed_scen_target(t,i,EAT_staples) = (
          i15_intake_scen_target(t,i) - sum(EAT_nonstaples,i15_intake_EATLancet(i,EAT_nonstaples)) )*(
          i15_intake_EATLancet(i,EAT_staples)/sum(EAT_staples2,i15_intake_EATLancet(i,EAT_staples2)) );

Now, the calorie supply at household level is calculated by multiplying daily per capita calorie intake with a ratio of supply to intake (f15_overcons_FAOwaste(i,kfo)) based on FAO estimates on historical food waste at consumption level and food conversion factors, and with a calibration factor f15_calib_fsupply(i). Another multiplicative factor accounts for increases in food waste over time.

    i15_kcal_pc_scen_target(t,i,kfo) = (f15_calib_fsupply(i)*f15_overcons_FAOwaste(i,kfo)
                                    *p15_intake_detailed_scen_target(t,i,kfo))
                                    *p15_foodwaste_growth(t,i);

In the last step, the regression-based calculation of daily per capita food demand is faded into the exogenous diet scenario according to a predefined spped of convergence:

    p15_kcal_pc_calibrated_orig(t,i,kfo) = p15_kcal_pc_calibrated(t,i,kfo);
    p15_kcal_pc_calibrated(t,i,kfo) = p15_kcal_pc_calibrated_orig(t,i,kfo) * (1-i15_exo_foodscen_fader(t,i))
                        + i15_kcal_pc_scen_target(t,i,kfo) * i15_exo_foodscen_fader(t,i);
);

The calibration parameter is added to the regression value. The BMI shares are not allowed to exceed the bounds 0 and 1. Values are corrected to the bounds. In case that the bmi groups, due to calibration, exceed 1, we rescale to 1. The mismatch below one is balanced by moving the exceeding quantities into the middle BMI group. We recalculate the intake with the new values.

After each execution of the food demand model, the body height distribution of the population is estimated. The starting point is the body height distribution of the last timestep. The body height estimates of the old period are moved into the subsequent age class (e.g. the 20-24 year old are now 25-29 years old). The age class of 15-19 year old is estimated newly using the body height regressions and the food consumption of the last 15 years.

   p15_bodyheight(t,iso,sex,"15--19","final") =
                     f15_bodyheight_regr_paras(sex,"slope")*
                     (sum(underaged15,
                       p15_kcal_growth_food(t,iso,underaged15)
                     )/3)**f15_bodyheight_regr_paras(sex,"exponent")
                     ;

The bodyheight of the underaged age class (0-14) is assumed to diverge from ‘normal’ body height by the same proportion as the age class of the 15-19 year old.

p15_bodyheight(t,iso,"M","0--4","final")=p15_bodyheight(t,iso,"M","15--19","final")/176*92;
p15_bodyheight(t,iso,"M","5--9","final")=p15_bodyheight(t,iso,"M","15--19","final")/176*125;
p15_bodyheight(t,iso,"M","10--14","final")=p15_bodyheight(t,iso,"M","15--19","final")/176*152;
p15_bodyheight(t,iso,"F","0--4","final")=p15_bodyheight(t,iso,"M","15--19","final")/163*91;
p15_bodyheight(t,iso,"F","5--9","final")=p15_bodyheight(t,iso,"M","15--19","final")/163*124;
p15_bodyheight(t,iso,"F","10--14","final")=p15_bodyheight(t,iso,"M","15--19","final")/163*154;

Finally, the regression outcome is calibrated by a country-specific additive term, which is the residual of the regression fit and observation of the last historical time step.

if (sum(sameas(t_past,t),1) = 1,
  p15_bodyheight_calib(t,iso,sex,age_new_estimated15) = f15_bodyheight(t,iso,sex,age_new_estimated15) - p15_bodyheight(t,iso,sex,age_new_estimated15,"final");
  p15_bodyheight(t,iso,sex,age_new_estimated15,"final") = f15_bodyheight(t,iso,sex,age_new_estimated15);
else
  p15_bodyheight_calib(t,iso,sex,age_new_estimated15)=p15_bodyheight_calib(t-1,iso,sex,age_new_estimated15);
  p15_bodyheight(t,iso,sex,age_new_estimated15,"final")=p15_bodyheight(t,iso,sex,age_new_estimated15,"final")+p15_bodyheight_calib(t,iso,sex,age_new_estimated15)*s15_calibrate;
);

Limitations There are no known limitations.

Definitions

Objects

Sets

Authors

Benjamin Leon Bodirsky, Isabelle Weindl, Jan Philipp Dietrich

See Also

09_drivers, 16_demand, 20_processing, 62_material, 70_livestock

References