Dimensionality and Functional Forms

The dimensionality of a model varies with the question to be answered. Typically, models built to illustrate theoretical propositions are parsimonious, capturing only the relevant economic relationships. Models designed to shed light on specific policy

questions for real economies are more detailed and their complexity reflects the nature o f the specific policy questions to be analyzed. Hence, a model that addresses investment policy will typically include an intertemporal representation, whereas a model that explores interhousehold tax incidence effects requires explicit representations of household types. In such models, domestic structures are emphasized, while the rest of the world is typically presented as a single agent. Conversely, models that focus on trade policies have explicit representations of several countries or trading regions, but employ simple structures to represent their domestic economies. Model detail often centres on the sectors and agents most likely to be affected by the policy change under question, while the remainder of the economy is modelled at a relatively more aggregate level.

The dimensionality of a model is limited by data availability, since results from a model that identifies agents or sectors for which no data exist are not credible. Paradoxically, an abundance of data can also influence dimensionality, as modellers face pressure from policy makers to include economic detail because it is available and, hence, is thought to make the model more realistic. Models with too much detail, however, impede an understanding of economic processes that drive the model results. Highly detailed applied general equilibrium models have developed reputations as black boxes into which a policy change is fed as an input and from which a set of results emerges with little explanation. In these models, the interactions that drive the model results can easily become obscured. Hence, the modeller's challenge is to balance clarity with realism in the presence of data constraints.

Once the dimensionality has been determined, modellers must also specify functional forms for the behavioural relationships in the model. They typically employ the family of 'convenient' functional forms for which the solutions to optimization problems can be obtained analytically. Cobb-Douglas and CES functions are widely used. Cobb-Douglas functions are simple, but highly restrictive, since they imply unitary income and uncompensated own-price elasticities, and zero uncompensated cross-price elasticities. In contrast, CES functions relax the unitary uncompensated own-price and zero cross-price elasticities of the Cobb-Douglas functions, but do so only by adding an additional parameter - the elasticity of substitution. Modellers often have information about the structure of an economy, such as literature-based elasticity estimates, which they wish to include in their model calibration. To incorporate this information, extra parameters are often injected into the model using nested CES functions, where the elasticity parameters enter at the various nests in the structure.

However, both CES and Cobb-Douglas preferences are homothetic and so yield demand functions which have unitary income elasticities. If income elasticities are thought to be significantly different from unity, some other functional form is needed, and a Stone-Geary/Linear Expenditure System with a displaced origin for utility measurement is commonly used. The minimum consumption requirements in such a system, which can be combined with either Cobb-Douglas or CES, are typically calibrated so that they reproduce literature estimates of income elasticities of demand in the neighbourhood of the base case equilibrium.

Some modellers have moved beyond this broad class of convenient functional forms to use variants of flexible functional forms, typically trans-log. The basis for

rejecting the convenient forms lies in the empirical results of econometric studies which reject the separability implicit in Cobb-Douglas and CES functions. The major drawback to using more flexible functional forms is that they are not always globally convex. Because the policy changes analyzed in many models can lead to a counterfactual equilibrium that is far from the initial equilibrium, the use of globally convex functions is often necessary to compute a model solution.

As with the choice of model structure, the functional forms used in a model should be attuned to the issue under investigation. Consider, for example, a trade model which explores the claimed long-term decline in the terms of trade of commodity­ exporting developing countries, and builds on the argument from Prebisch (1962) and Singer (1950) that developing countries export necessities and import luxuries, such as capital goods. Such a model requires income elasticities of demands that are different from unity to reflect the feature that growth in both the developed and the developing countries will adversely affect the developing country's terms of trade. This feature emerges if the income elasticities of import demand in developed countries are less than one while those in developing countries are greater than one. Using models with either Cobb-Douglas or CES preferences will not meet these conditions and a different functional form is needed. On the other hand, if the income effects from the change considered in the model are thought to be small compared to the relative price effects, a model with homothetic preferences may suffice.

As with the choice of model structure, decisions about dimensionality and functional forms are dependent on the research question. Both sets of decisions need to

balance simplicity with realism; but other pragmatic considerations such as computational feasibility also enter the choice.