Leontief Production Economic Utility

Wassily Leontief Aggregation, Leontief Production and Utility

We have already made reference to the problem of aggregation in discussing the input-output model. In the model, the in­numerable separate activities which make up the economy are aggregated into a number of sectors, the flow of products between which is then analysed. In other words, some kind of consolid­ation into sectors is required. A number of authors have contributed to the literature on this topic (see Green, 1964, Chap. 9). Leontief's own contribution to aggregation theory, which dates back to some early work on index numbers in the 1930s, was more fundamental. He was concerned with the following sort of problem. A functional relationship exists between a number of variables. In what circumstances are we justified in replacing that functional relationship with another in a small number of variables, where one or more of the new variables is found by aggregating or consolidating several of the original variables into a single number? The problem arises frequently in economics. We might wish to aggregate a number of commodities in a utility function into a single commodity group or to aggregate the inputs of separate kinds of capital services into a single capital aggregate in a production function.

His two 1947 papers on aggregation set out what became known as the Leontief condition for aggregation (Leontief, 1966b, Chap. 13). It states essentially that aggregation of two variables in the original function is permissible if and only if the marginal rate of substitution between the two variables depends only on those two variables, and is unaffected by changes in variables outside the group. For example, two food products can be aggregated into a single variable 'food' in a utility function, if and only if the marginal rate of substitution between them in consumption is unaffected by the quantities consumed of, say, housing or shirts. The inputs of two kinds of capital services can be aggregated into a single variable, 'capital', in a production function, if and only if the ratio of their marginal products is independent of the input of labour (Bliss, 1975, Chap. 7).

The implications of this result are substantial. On one hand it has stimulated demand theorists such as Gorman and Strotz to investigate preference structures which yield consumer demand functions in terms of commodity groups rather than individual commodities. On the other hand the Leontief condition, by imposing restrictive conditions for consistent aggregation, has cast serious doubt on the 'neo-classical parables' which seek to explain the distribution of income on the basis of production functions in which capital appears as a single aggregate. It has always been clear to Leontief that the Cambridge criticisms of simple neo-classical capital and distribution theory are well-founded.