Volume preserving CES and CET formulations

Two of the most widely used functional forms in quantitative economic analysis are the constant-elasticity-of-substitution (CES) and constant-elasticity-of-transformation (CET) functions. The CES functional form is oft-used to represent production functions--for example, in macro growth models or in multi-sector CGE models--or preference functions--for example the Armington specification for allocating domestic absorption across domestic and imported goods. In other contexts, the CET functional form is used to allocate production or supply across different markets--for example land-use. As economic models increasingly integrate bio-physical properties, one potential drawback of the CES/CET is that they do not preserve additivity, i.e. the sum of the volume components do not add up to the total volume. The volume discrepancies can potentially be small, though a paper by Fujimori et al 2014 suggest that in the case of long-term land allocation, the discrepancies can be large, particularly at the more (spatially) disaggregated level. Even if the volume discrepancies are small, deviations from strict additivity are often confusing to scientific colleagues from other disciplines. This paper explores the use of a modified CES/CET function that has virtually the same attributes of the standard CES/CET function but preserves additivity. The additive form of the CET has been introduced in labor allocation decisions by Dixon and Rimmer 2003/2006 and land-use allocation by Giesecke et al 2013. This paper provides the analytical formulation of both forms of the CES/CET function. It compares the implementation of the additive CET using a small numerical example to highlight some key potential differences. It then introduces the additive form of the CET function in the land-use specification of the Envisage Model and assesses the impacts of the two formulations on agricultural output, prices and land-use in a long-term global scenario.