*■1
then having an aggregate production function is simple because all outputs and inputs are homogeneous and mobile between firms so that each factor input o f the same kind can be substituted for the other o f the same sort In other words, the marginal rates of technical substitution among the inputs o f similar kinds are constant and the marginal factor productivity is identical (Star 1974). The aggregate production function can therefore be written as Y = F (Z, N) (A3.5) where ZYi = E /'(Z \ N1) (A3.6) ENj i~i (A3.7) S z , i»7 (A3.8)
Since a production function describes the maximum level of output that can be so achieved if the inputs are effectively employed (Fisher 1969b), it is implicit here that the individual allocations of labour and capital to firms, as they are homogeneous and mobile, will be determined in the course of the maximization problem (May 1946,
1947).
Heterogeneous capital: Suppose that capital differs from firm to firm and that capital of vintage (type) i can be denoted by Kj. Implicit here is that technical changes are embodied in the capital inputs. Only labour is now mobile and can be allocated to firms to maximise total output while capital is assumed to be firm-specific.
Since each capital may have different technical properties, each firm's production function will, in general, be different from one another. That is
Y, = / l (Zl, N i) (A3.9)
In this case, Nataf (1948) showed that an aggregate production function o f the form Y = F (Z, N) (A3.10) where Y = E Yj (A3.12) .=/ N = X N ; (A3.13) h r !
Z = Z (Zj, Z2, ..., ZJ is a capital aggregate independent of labour N.
will exist if and only if every firm's production technology is additively separable in capital and labour. That is, if and only i/each firm's production function A3.10 can be written in the form
Y, = /•■ (Z„ N,) = z f (Z) + m i (Nt) (A3.14)
i = 1, 2 ,..., n
However, according to (Fisher 1969b), additive separability is a sufficient condition for capital aggregation whether or not labour is optimally allocated to firms.
So going back to the conditions under which an aggregate production function exists, the Leontief Theorem says, a necessary and sufficient condition for the existence o f such an aggregate production function is that the marginal rate of technical substitution between any pair of Z, in the production of the maximum Y is independent o f N (Solow 1956b and Fisher 1969b). Fisher noted the implication of this condition
about the original firm production functions. That is, if we assume strict diminishing returns to labour (f1^ < 0) then it can be shown that a necessary and sufficient condition for capital aggregation is that every firm’s production function satisfies a partial differential equation in the form
if*zn) / i f *nn) = £ < / * ) (A3.15)
where function g is the same for all firms.
We can see from equation A3.15 that if / ‘zn= 0, we can have a capital aggregate as Nataf (1948) pointed out because the implication of / ‘zn = 0 is that each of the firms’ production function is additively separable. Fisher (1965) acknowledged that since additive separability is not a very reasonable property of production functions because if one or two firms' production functions do not have the additive separability property, then it is not possible to have a capital aggregate. We may discard the possibility that /
'zn = 0 •
Heterogeneous capital under constant returns to scale: Suppose that each firm’s production function exhibits constant returns to scale, while labour and output are homogeneous but each unit of capital varies. Fisher (1965) theorises that in this case a necessary and sufficient condition for the existence of an aggregate capital stock is that "when labour has been optimally allocated so that its marginal product is the same for all vintage of capital, the average product of labour shall be the same for all vintages". Mathematically, a function g independent of the number of firms i is such that for all i =
1, 2 ,.., n
/■7N, = (Z ,/'z) / N , + / -n = Fisher (1969) provides the proof for this condition.
Heterogeneous capital under capital-augmenting technical differences and constant returns to scale: Suppose that each firm's production functions differ from each other only by a capital-augmenting technical difference. This is a slightly different generalisation of the above case. Each firm's production function can be written as
/* (Z,, N,) r / i ( b £ . Ni) (A3.17)
where
i = 1, 2,.., n
bb b2, ..., b„ are set of positive constants
Intuitively, if technical change is capital-augmenting, one unit of the new capital equipment is worth a constant number of units of the old capital input. Fisher (1965) theorises that in this case, an aggregate capital stock
Z = h (Z 1,Z 2,..,Z n) (A3.18)
exists "if all technical change is capital-augmenting". The proof is given by Solow (1964) and Fisher (1965).
Heterogeneous capital under capital-altering technical difference and non-constant returns to scale: Without constant returns to scale, there is a limited class of cases where capital aggregation is possible. This is the class in which each firm's production function can be made constant returns to scale after a suitable stretching of the capital axis (Fisher 1969). Fisher refers to this class as capital-generalised constant returns to scale (CGCR). This is a generalised case of capital-augmenting technical differences. Fisher also refers to this generalised case as capital-altering technical differences.
If each firm's production function differ from each other only by a capital- altering technical difference then its production function can be written as
/ * ( Zj,N i ) = F i [ / / i ( Z i ) , Ni] (A3.19) where