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

Increases in the level of aggregate output (utility) with relative input (commodity)
price ratios fixed has no affect on factor intensity since the expansion path is a straight line
from the origin. 20 Moreover, the assumption of weak separability ensures that the
introduction of other inputs (commodities) not in the aggregator function also has no
consequence for factor (commodity) usage ratios. Hence, changes in input (commodity)
intensities xi will only be a function of the relative prices of various types of input xi in that
part of the nest.
Allanson (1989) also notes that relative price changes in one nest can have indirect
effects on input (commodity) allocations elsewhere in the nest. Referring to the nested
structure in Figure A1, if the price of input x2 increases, this will affect the optimal
combination of x1 and x2 in the aggregate nest, but due to the separability restriction, it will
not directly affect the optimal use of x3. There will, however, be an indirect effect on the use
of x3 due to a rise in the composite price of aggregate input X. This implies that the firm will
substitute x3 for aggregate X in the top nest. Moreover, if x3 was an aggregate input, then as
a consequence of linear homogeneity, its increased use would be translated proportionally
to all inputs in that nest.
Thus, if expression (A.1) satisfies both weak separability and linear homogeneity,
then the underlying production function is said to be weakly homothetically separable (or
'homogenously separable' Green 1971, pp.152-156) and ensures consistent aggregation. 21
Consistent aggregation makes it possible to index correctly over prices and quantities when
forming composites such that multi-stage nested optimisation procedures give equivalent
results to single stage optimisation problems (Ozanne, 1992).
20 Increases in aggregate output (utility) are movements onto higher isoquants (indifference) curves;
Expansion paths join points of cost minimising equilibria.
21 It is important to note that weak homothetic separability does not imply that the production function itself
is homothetic.
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