1 A Recursive Dynamic Computable General Equilibrium Model For ...

Appendix A: Nesting
The choice of function under conditions of model calibration favours the use of
simpler 'convenient functional forms'. The drawback, however, is that simpler functional
forms greatly restrict the number of parameters within the function, which in turn inhibits
the degree of flexibility when characterising producer/consumer behaviour.
A common response to this problem is to employ a separable nested (or hierarchical)
structure, whereby an assumption is made about the partitioning of the elements of the
underlying production/utility function into different groups and aggregations. Hence, the
assumption of separability implies that constrained optimisation is undertaken in several
stages. Nested structures then allow a greater number of elasticity parameters at each stage
of the production/utility function. This increases the flexibility of the model, without
burdening computational facility.
Separability and Aggregation
In order to undertake a two-stage nested optimisation procedure, two conditions
must be met. First, to permit a partitioning of the inputs, Strotz (1957) devised the concept
of weak separability. A precise definition of separability is given by Chambers (1988) who
notes,
'separability hinges on how the marginal rate of technical substitution (MRTS) between two inputs responds
to changes in another input' (pp.42).
To illustrate the relationship between separability and multi-stage optimisation, a theoretical
example is employed. Assume a 3 factor (xi i=1,2,3) production function which is of the
form: 19
Y = f ( X , x3
)
(A.1)
where input X is represented as an aggregator function consisting of inputs x1 and x2:
, ) x g X = (A.2)
( 1 2 x
A schematic representation of this two-level nested structure is presented in figure 2.5.
19 This theory is equally applicable to the utility function in consumer theory.
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