Separability

‘Suppose you were interested in the demand for tomatoes in Ireland’ (Gorman 1987: 305). Thus, begins Gorman’s article on separability in the Palgrave Dictionary of Economics, recalling his own early applied work, characterised by his widow Dorinda as involving ‘careering around Dublin on a bike, looking in greengrocers’ windows’ (private communication).

As for the researcher, so also for the household or the enterprise. Practical decision-making often calls for shortcuts relative to full intertemporal optimi­sation of a preference function. Gorman was confident that in reality most households engage in two-stage budgeting, in which the family budget is first allocated between broad classes of spending (clothing, food, etc.) and then choices are made within each class.

How good is this as a way of making decisions? Each of the two stages is problematic. Can the first-stage allocation safely be made just on the basis of some price aggregates for each class of goods, and without looking at the relative prices of all goods? Even if the first-stage allocations are correct, can the choice of goods within each class safely be made without reference to the prices on offer or quantities chosen of goods in other classes? It turns out that the validity of such a procedure for achieving the optimum requires that the household’s utility function satisfy some fairly drastic separability restric­tions—more stringent than had been recognised in the literature.

In particular, Strotz (1957) had argued that a sufficient condition for two- stage budgeting is that the household’s utility function be separable, that is, expressible in the form:

where x_r denotes the vector of consumption in class r. Gorman showed that, while necessary, separability is not sufficient.^[160] In addition, it is required that the sub-utility functions, which Gorman called “specific satisfaction func­tions”, v_r, enter utility either additively or through an intermediate function which is homogeneous of degree one in its components.^[161]

That these constraints were severe was for Gorman ‘in a sense a good thing’ (Gorman 1959a: 475); since (he knew) households did adopt two-stage bud­geting, it must be that their preferences were so restricted. Knowledge of this fact would ease the task of applied researchers wishing to estimate the relevant parameters.

What motivated Gorman here was the tension between two goals of eco­nomic modelling. On the one hand, the conceptual need for a coherent and psychologically or organisationally credible theoretical representation of decision-making; on the other hand, the operational need to have a workable algebraic representation of this behaviour. The basic assumptions of utility theory are too weak to yield specific functional forms or to make many pre­dictions about individual or aggregate behaviour. Further assumptions are needed if real progress is to be made in applied economics, but these assump­tions must be more-or-less reasonable. Looking from the other side, it is evi­dent that simple algebraic representations of behaviour are needed for applied econometrics. Simplicity is also needed if the theory is to be mathematically manipulated to yield further predictions. But all such uses are empty if the algebraic specification implies incoherent decision-making. In practice, most of the algebraic representations with which demand and production theory deal are linear functions of prices or quantities, or are simple transformations of linear functions.

An interesting example of how specific separability assumptions could help in underpinning a linear representation of behaviour is provided by Gorman's 1982 paper “Facing an Uncertain Future”. In this paper, Gorman's goal is to show that the assumptions required to justify a linear representation of the intertemporal objective function are much weaker and more credible than had hitherto been recognised in the literature.

For a static environment, Allais, Samuelson, Von Neumann and Morgenstern and others had presented the conditions under which decision­making under conditions of uncertainty could be represented as the maximi­sation of a linear function—a weighted average—of the various alternative possibilities.^[162] The key assumption in this expected utility hypothesis, Samuelson's weak independence axiom (or “sure thing principle”), is one of separability.

If we widen the focus to intertemporal decisions (still under uncertainty), can we get as simple an objective function with equally weak assumptions? The objective function that is commonly—indeed almost universally used— is a double sum:

where y_st is the vector of flows which occur in period t if state s occurs.^[163] Can we derive such a simple form from assumptions that are as mild and accept­able as those underlying expected utility? If we are prepared to assume an extended version of the sure thing principle, so that it applies over time as well as between uncertain states of the world, we will get this double summation

form of the objective function. But Gorman points out that extending weak independence in this way is logically problematic.

Before doing so, he notes that such an extension to a second dimension is permissible in the case of a social welfare function under uncertainty, where households rather than time are the extra dimension.

However, to assume that households or firms are not only able to calculate their utility over all possible future states of the world but assert independence over each set of states of the world and time periods is a step too far for Gorman. Such an argument ‘assumes from the outset that we are all very bright, and especially so at computation’ (ibid.: 214). Instead, he proposes the contrary idea, that ‘we are all pretty limited beings, only able to hold a few things in our minds at a time...and that organisations are collectively quite as limited as their members’ (ibid.). Specifically, he assumes that ‘we look ahead two periods in detail, summarising the impact of our choices on more distant prospects in a single figure’ (ibid.). He then proceeds to show that this, par­tially myopic but more realistic, vision of decision-making, embodying a very weak (undemanding) form of intertemporal separability, is enough to gener­ate the double summation form of the objective function.

Here, Gorman has armed applied econometricians with a justification for doing what they had always intended—use a linear functional form. The behavioural assumptions are somewhat restrictive, but also characteristically down-home: the firm is planning for now and next year, and for a general sense of what it will bequeath in later years. If that is not how firms and households behave exactly, it seems, at the same time, to be not too unrealistic.

3.3