Using CRETH to make quantities add up without efficiency bias

Modern CGE models can boast considerable sectoral detail. However, it is obvious that output of (say) electronic components, must be quite heterogeneous. Hence, since Leontief, multisectoral models tend to measure quantities not in physical units but in effective economic units (usually initial-dollars-worth). The CET functional form, close cousin to CES, is used to allocate a fixed resource between alternate uses; for example land between crops, or workers between sectors. It works well when both input and output quantities are measured in initial-dollars-worth, such as land rental values. Because CET chooses a crop mix to maximize revenue, it is welfare-neutral -- a small change in land allocation will not affect land's contribution to GDP. This is a desirable property. But CET translates poorly into physical units: we typically find that if percent changes in (effective) land use are interpreted as percent changes in crop areas, then total land area is not fixed. This can be a problem for reporting results, or for interfacing a CGE model to ecological or agronomic models which work with physical units. The CRETH functional form is a generalization of CET that has in the past been used like CET to allocate a fixed (measured in effective units) resource between alternate uses. In this usage, CRETH is like CET, but with more parameter flexibility. Here we show that CRETH land supply functions can instead be interpreted in a more literal fashion: as the answer (FOC) to a revenue-maximizing problem, where a land-owner allocates a fixed acreage of land between uses. Used in this way, CRETH (a) allows reported land areas to add up properly, and (b) has the optimum property that small changes in land allocation do not affect the land contribution to GDP (so avoiding efficiency bias).