Modelling

Box A1: The Cobb-Douglas production function in T21 for agriculture, industry and services macro sectors

The classic form of the CD production function is expressed as following: Y = A × Kα

× L(1-α)

Where here the traditional technology term, A, is used to represent a series of factors affecting total factor productivity (TFP; as in a growth accounting approach), K represents the stock of capital, and L represents labour. The constant α represents the elasticity of output to capital: the ratio between the percentage change of output and the percentage change of an input. The elasticity of output to labour is set to 1-α, assuming that there are constant returns to scale (the production function is thus first order, homogeneous). In T21 the standard CD production function is transformed into a more transparent algebraic form, and TFP is expanded to include several different elements.

The equation used to estimate industry production is as shown below: yit

= yi0 Where yit × rict α × rilt (relative to 1970), ricl β × fpit is the current industry production, yi0 is the initial industry production, rict is the relative industry labour and fpit is the relative industry capital is the industry total factor productivity. Moreover, α is

the elasticity of capital and β is the elasticity of labour. The T21 takes an approach whereby total factor productivity is comprised of a number of components related to human and natural capital. Thus, total factor productivity in industry fpit

life expectancy rlet rate rwrt

), energy (relative oil price ropt ), relative waste recycle , and relative water stress rwst . The total factor productivity of industry is calculated as follows, with relative

oil price and water stress having a negative impact on productivity, reflecting the negative effects that scarcities of these have on industrial production, either through higher prices or other costs that have to be incurred to compensate:

ƒpit = ryst α / ropt c × rlet β × rwrt d × rwst e

The equation used to estimate agricultural production is defined in terms of yield, still determined by a transformed Cobb-Douglas production function, uses different inputs for TFP. The equation below is used to estimate natural yield per hectare. Effective crop yield is the natural crop yield per hectare minus yield lost due to pest diseases. By multiplying the harvested area by effective crop yield per hectare, we determine the total crop yield. Total crop yield multiplied by crop value added gives agriculture (food processing) production, or the total value added.

yt Where yt = yit-1

capital, and rlt soil quality), ƒt

× rct α × rlt β × ƒ(R &#38; D, sq, ƒt, , 1/ws) is the current natural crop yield per hectare, yit-1 is the initial natural crop yield per hectare, rct is the relative

is the relative labour. Where ƒ is the effect of R &#38; D (relative research and development), sq (relative (relative fertiliser use) and ws (relative water stress) on crop yield. Moreover, α is the elasticity of

capital and β is the elasticity of labour. Labour in the agriculture production function represents human capital that consists of quantity and quality of labour. The quantity of labour is agriculture employment while quality of labour is determined by literacy (average years of schooling) of the labour force and health conditions (life expectancy).

selected variables (e.g. availability of reserves and resources, or the elasticity of GDP to oil prices). Further, extreme condition tests,

feedback loop analysis as

well as unit consistency tests were performed on all models. Further, boundaries as well as structural (i.e. causal relations and equations) and parameter consistency tests were normally checked with experts in the field analysed. Overall, the structure of the models presented in the five studies presents less detailed disaggregation but higher dynamic complexity (cross sectoral relationships and feedback loops) when compared with other existing models (e.g. MARKAL,

in the energy sector). In other words, each sector developed for the studies is relatively simple when taken in isolation, and the complexity comes out of the feedback loops built into the model across modules and sectors.

Concerning behavioural validation, over 450 social, economic and environmental variables were simulated against history. Historical projections generally match well with data, as partly illustrated in Figures 5, 6, 8 and 9, and, as shown in section III in the Technical Background Material. During the modelling process particular

539

is determined by a range of human and natural capital related components, including health (relative ), education (relative years of schooling ryst

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