# Sandbox

$ AGP_{r,f=1} = YL_{r} * LD_{r,l=1} $

$ CD_{r}=\frac{YL_{r,t=1}}{{KAG_{r,t=1}^{CDALF_{r,s=1}}}*{LABS_{r,S=1,t=1}^{(1-CDALF_{r,s=1})}}} $

## Crop Production

Crop production (AGP_{f=1}) i is the product of yield and land devoted to crops (LD_{l=1}).

- $ AGP_{r,f=1} = YL_{r} * LD_{r,l=1} $

We focus here on the determination of yield; the amount of land devoted to crops is addressed in the Land Dynamics section.

Yield functions are almost invariably some kind of saturating exponential that represents decreasing marginal returns on inputs such as fertilizer or farm machinery. Such functions have been used, for instance in World 3 (Meadows, 1974), SARUM, (SARU, 1977), the Bariloche Model (Herrera, et al., 1976), and AGRIMOD (Levis, et al., 1977). IFs also uses a saturating exponential, but relies on a Cobb-Douglas form. The Cobb-Douglas function is used in part to maintain symmetry with the economic model but more fundamentally to introduce labor as a factor of production. Especially in less developed countries (LDCs) where a rural labor surplus exists, there is little question that labor, and especially labor efficiency improvement, can be an important production factor.

In the first year of the model, yield (YL) is computed simply as the ratio of crop production (AGP_{f=1}) to cropland (LD_{l=1}). The determinations of the initial values of these variables are described elsewhere in this document. It is bound, however, to be no greater than 20 tons per hectare in any country.

In forecast years, IFs computes yield in stages. The first provides a basic yield (BYL) representing change in long-term factors such as capital, labor and technology. The second stage uses this basic yield as an input and modifies it based on prices, so as to represent changes in shorter-term factors (e.g. amounts of fertilizer used, even the percentage of land actually under cultivation). Finally, in a third stage, yields are adjusted in response to changing climate conditions.

The basic yield (byl) relates yield to agriculture capital (KAG), agricultural labor (LABS), technological advance (AGTEC), a scaling parameter (CD), an exponent (CDALF), and a saturation coefficient (SATK).

- $ byl_{r}=cD_{r}*{(1+AGTEC_{r})}_{t-1}*KAG_{r}^{CDALF_{r,s=1}}*LABS_{r,s=1}^{{(1-CDALF}_{r,s=1)}}*SATK_{r} $

The equations for KAG and LABS are described elsewhere (see the Capital Dynamics and the economic model, respectively).

cD is a scaling factor calculated in the first year of the model based upon the base year yield (YL), capital (KAG), and labor supply (LABS). It is similar to the shift factors elsewhere in the model, which are used to match predicted values in the base year to actual values. It does not change over time.

- $ CD_{r}=\frac{YL_{r,t=1}}{{KAG_{r,t=1}^{CDALF_{r,s=1}}}*{LABS_{r,S=1,t=1}^{(1-CDALF_{r,s=1})}}} $

CDALF is the standard Cobb-Douglas alpha reflecting the relative elasticities of yield to capital and labor. It is computed each year in a function, rooted in data on factor shares from the Global Trade and Analysis Project, driven by GDP per capita at PPP.

AGTEC is a factor-neutral technological progress coefficient similar to a multifactor productivity coefficient. It is initially set to 1 and changes each year based upon a technological growth rate (TECHGROAG).

- $ AGTECH_{r}=AGTECH_{r,t-1}*(1+TECHGROAG_{r}) $

The algorithmic structure for computing the annual values of TECHGRO involves four elements:

The difference between a targeted yield growth calculated the first year and the portion of that growth not initially related to growth of capital and labor (hence the underlying initial technology element of agricultural production growth); call it AGTECHINIT. This element is assumed to decrease by half over 100 years.

The gap between desired global crop stock levels and actual stocks (hence the global pressure for technological advance in agriculture); call it AGTECHPRESS. This contribution is introduced by way of the ADJUSTR function of IFs.^{[1]}

The difference between the productivity of the agricultural sector calculated in the economic model and the initial year's value of that (hence reflecting changes in the contributions of human, social, physical, and knowledge capital to technological advance of the society generally); call if AGMFPLT.

The degree to which crop production is approaching upper limits of potential; this again involves the saturation coefficient (SATK).

The algorithmic structure this is:

- $ TECHGROAG_{r}=F(AGTECHINIT_{r},AGTECHPRESS_r,AGMFPLT_{r}, SATK_{r}) $

The saturation coefficient is a multiplier of the Cobb-Douglas function and of the technological change element. It is the ratio of the gap between a maximum possible yield (YLLim) and a moving average of yields to the gap between a maximum possible yield and the initial yield, raised to an exogenous yield exponent (** ylexp **). With positive parameters the form produces decreasing marginal returns.

- $ SATK_{r+1}=(\frac{YLLim_{r}-syl_{r}}{YLLim_{r}-YL_{r,t=1}})^{ylexp} $

*where*

syl_{r} is a moving average of byl, the historical component of which is weighted by 1 minus the user-controlled global parameter ** ylhw **.

** ylexp ** global parameter

The maximum possible yield (YLLim) is estimated for each country and can change over time. It is calculated as the maximum of 1.5 times the initial yield (YL_{r,t=1}) and the multiple of an external user-controlled parameter (** ylmax **) and an adjustment factor (YLMaxM).

- $ YLLim_{r}=max (\mathbf{ylmax}_{r}*YLMaxM_{r},1.5*YL_{r,t=1}) $

*where*

** ylmax ** is a country-specific parameter

The adjustment factor YLMaxM allows for some additional growth in the yields for poorer countries

- $ YLMaxM_{r}=1*(1-DevWeight_{r})+((\frac{YL_{r}}{YlMaxFound})^{0.35}*DevWeight_{r}) $

*where*

DevWeight_{r} is the GDPPCP_{r}/30, with a maximum value of 1

YlMaxFound is the maximum value of YL found in the first year

Before moving to the next stage, a check is made to see if the growth in byl is within reason. Specifically, byl is not allowed to exceed the moving average of byl (syl) times a given growth rate (ylgrbound). This bound is the maximum of a user-controlled global parameter - ** ylmaxgr ** and an initial country specific target growth rate (tgryli

_{r}).

^{[2]}This latter target growth rate in yield is set in the first year and is a function of current crop demand (AGDEM), expected crop demand (etdem), and a target growth rate in cropland.

- $ tgryli_{r}=\frac{etdem}{AGDEM_{r,s=1}}-1-\mathbf{tgrld}_{r} $

*where*

** tgrld ** is a country-specific parameter indicating target growth in crop land

etdem is an initial year estimate of the sum of industrial, feed and food demand for crops in the following year

At this point, the basic yield (BYL) is adjusted by a number of factors. The first of these is a simple country-specific user-controlled multiplier – ** ylm **.

**can be used to represent the effects of any number of exogenous factors, such as political/social management (e.g., collectivization of agriculture).**

*ylm*The basic yield represents the long-term tendency in yield but agricultural production levels are quite responsive to short-term factors such as fertilizer use levels and intensity of cultivation. Those short-term factors under farmer control (therefore excluding weather) depend in turn on prices, or more specifically on the profit (FPROFITR) that the farmer expects. Because of computational sequence, we use domestic food stocks as a proxy for profit level.

In this second stage, the recomputed yield (YL) is multiplied by a stock adjustment factor.

- $ YL_{r}=BYL_{r}*stockadjustmentfactor_{r,f=2}(\mathbf{elfdpr1,elfdpr2}) $

The stock adjustment factor uses the ADJSTR function to calculate an adjustment factor related to the current stocks, the recent change in stocks, and a desired stock level. The desired stock level is given as a fraction (agdstl) of the sum of crop demand (AGDEM_{f=1}) and crop production (AGP_{f=1}). agdstl is set to be 1.5 times ** dstl **, which is a global parameter that can be adjusted by the user.

The focus in IFs on yield response to prices differs somewhat from the normal use of price elasticities of supply. For reference, Rosegrant, Agcaoili-Sombila, and Perez (1995: 5) report that price elasticities for crops are quite small, in the range of .05 to .4.

In the third stage. IFs considers the potential effects of a changing climate on crop yields. This is separated into two parts: the direct effect of atmospheric carbon dioxide concentrations and the effects of changes in temperature and precipitation.

The direct effect of atmospheric carbon dioxide assumes a linear relationship between changes in the atmospheric concentration from a base year of 1990 and the percentage change in crop yields.

- $ CO2Fert_{t+1}=\mathbf{envco2fert}*\frac{CO2PPM-CO2PPM_{t=1990}}{CO2PPM_{t=1990}} $

*where*

** envco2fert ** is a global, user-controllable parameter

CO2PPM_{t=1990} is hard coded as 354.19 parts per million

The effect of changes in annual average temperature and precipitation are based upon two assumptions: 1) there is an optimal temperature (Topt) for crop growth, with yields falling both below and above this temperature and 2) there is a logarithmic relationship between precipitation and crop yields. The choice of this functional form was informed by work reviewed in Cline (2007). Together, these result in the following equation:

- $ ClimateEffect_{t+1}=100*(\frac{e^{-0.5*\frac{(TO_{r}+DeltaT_{r}-Topt)^{2}}{SigmaTsqd}}*ln(P0_{r}*(\frac{DeltaP_{r}}{100}+1))}{e^{-0.5*\frac{(T0_{r}-Topt)^{2}}{SigmaTsqd}}*ln(P0_{r})}-1) $

*where*

T0 and P0 are country-specific annual average temperature (degrees C) and precipitation (mm/year) for the period 1980-99.

DeltaT and DeltaP are country specific changes in annual average temperature (degrees C) and precipitation (percent) compared to the period 1980-99. These are tied to global average temperature changes and described in the documentation of the IFs environment model.

Topt is the average annual temperature at which yield is maximized. It is hard coded with a value of 0.602 degrees C.

SigmaTsqd is a shape parameter determining how quickly yields decline when the temperature moves away from the optimum. It is hard coded with a value of 309.809.

CO2Fert and ClimateEffect are multiplied by each other to determine the effect on crop yields.

There are two final checks on crop yields. They are not allowed to be less than one-fifth of the estimate of basic yield (BYL) and they cannot exceed the country-specific maximum (** ylmax **) or 50 tons per hectare.

</references> The ADJSTR function, used throughout the model, is a PID controller that builds in some anticipatory and smoothing behavior to equilibrium processes by calculating an adjustment factor. It considers both the gap between the current value of the specific variable of interest, here crop stocks, and a target value, as well as change in the gap since the last time step. Two parameters control the degree to which these two "differences" affect the calculation of the adjustment factor. In this case, these are the global, user-controllable parameters ** elfdpr1 ** and

**.**

*elfdpr2***is reduced for countries with syl>5, falling to a value of 0.01 when syl>=8.**

*ylmaxgr*