## Aggregated Production Functions

However, there are a number of strong arguments against the use of production functions that we need to acknowledge and address if possible. The argument may be dated to the years immediately after World War II when economists were busy reconstructing historical statistics and national accounts, and the aggregate production function was in the process of being implemented as a practical tool.

The first question that arose was, not surprisingly, how the aggregate macroeconomic function should be related to the microeconomic production functions that characterize individual firms. There were two schools of thought with regard to this issue. Klein argued that the aggregate function should be strictly a technical relationship, comparable to firm-level production functions, and not reflecting behavioral assumptions such as profit maximizing (Klein 1946, p. 303, cited by Felipe and Fisher 2003):

There are certain equations in micro-economics that are independent of the equilibrium conditions and we should expect that the corresponding equations in macro-economics will also be independent of the equilibrium conditions. The principal equations that have this independence property are the technological production functions. The aggregate production function should not depend on profit maximization but purely on technical factors.

Klein's view would be consistent with that of Leontief (1941). However the 'technological' view was immediately disputed (for example, May 1947; also quoted by Felipe and Fisher 2003):

The aggregate production function is dependent on all the functions of the micro-model, including the behavior equations such as profit maximization, as well as all exogenous variables and parameters . . .

It will be clear in due course that the latter viewpoint has prevailed in the literature.

The next obvious problem was how to account for capital. Here again, two views emerged. One view, most strongly espoused by Joan Robinson at Cambridge (UK), was that capital stock should be measured in physical terms (Robinson 1953-4).This left open the question of how to measure heterogeneous physical capital stock in monetary terms. This question initiated the so-called 'Cambridge controversy' which has never really been resolved in the literature, notwithstanding Robinson's title-page assertion in 1971 (Robinson 1971). However, it has been resolved in the sense that the so-called 'perpetual inventory method' or PIM, developed especially by Angus Maddison, is now widely used in practice. This method measures capital stock as the accumulation of real (deflated) capital investment, less depreciation.1 The standard objection to this approach is that the monetary value of capital depends upon prices, which can change for reasons unrelated to productivity. For example, the costs of capital equipment clearly reflect energy (exergy) prices at the time of manufacture.

In this book, we propose a partial reconciliation of the physical interpretation of capital and the economic interpretation. In short, we can adopt Kümmel's view that capital equipment is 'productive' only insofar as it contributes directly or indirectly to the function of extracting exergy resources, transporting them, converting energy (exergy) into useful work and work products including information, or utilizing such products for purposes of subsistence or enjoyment (for example, Kümmel et al. 1985). Obviously some types of capital - notably engines and related machines - convert energy directly into work, or perform work on work-pieces that eventually become components of products, including machines. Other types of capital protect the machines, or the associated infrastructure. The point is that virtually all types of capital (economically speaking) are involved in the exergy-work-production-service function and can therefore by measured in terms of exergy embodiment or exergy consumption.

A related problem is the implicit assumption that only two, or three, independent variables can really account for the output of the economy, as a dependent variable, over periods. Furthermore, it is a fact that any smooth twice-differentiable function of several variables - whether homogeneous of degree one or not - implies that the function exists for all possible combinations of the arguments. Since any combination is possible, the implication is that the variables can be substituted for each other throughout their ranges. In the two-factor case, this means that a specified output can be obtained with infinitesimal labor if there is enough capital, or conversely, with infinitesimal capital, with enough labor. The introduction of a third factor does not affect this conclusion: it implies that economic output is possible without any input of X (energy or useful work). In short, an attribute common to all production function models is the built-in assumption of complete substitutability between all of the factors.

Difficulties with the assumption of substitutability were discussed at some length in the previous chapter. Indeed, we know that there are limits to substitutability. In fact, all three inputs to the current economy are essential, which means non-substitutable except at the margin and over time. It is the essentiality of certain inputs (not only capital, labor and exergy) that imposes a multi-sectoral structure on the real economy. This, in turn, makes the output elasticity of an essential input - whether it be food, fresh water, copper or petroleum - much greater than its apparent cost share in the national accounts.

Evidently, substitutability is a variable concept, depending on the time element. It is arguable that instantaneous (for example, overnight) substi-tutability is essentially null. The economy has a great deal of inertia and there is really no possibility of substituting labor for capital, or capital for useful work - or conversely - in the very short term. A theoretical distinction was made between the movement of firms along a production frontier, versus movement between production frontiers (Solow 1957). Instantaneous substitution of this kind (if it were possible) would correspond to movements along the production frontier. This would correspond to increasing capital intensity (or 'capital deepening') without techno logical change.

However, the production frontier moves outward to a new frontier due to the combined effect where new capital (machines) also incorporates technological improvements. The importance of embedding technological change in new capital equipment and 'learning by doing' was emphasized by Arrow (1962). There is an important asymmetry between the degree of choice (of techniques) available before and after new machines have been installed. The flexible situation before installation of new machines has been characterized as 'putty', while after the machines are in place it becomes 'clay' (for example, Fuss 1977). For a broad survey, see Baily et al. (1981). Applications to the specific case of energy use have been reviewed by Atkeson and Kehoe (1999).

Within the standard theory of growth, there is a range of specifications with regard to the relative importance of these two modes: 'pure capital deepening' versus 'pure technological advance'. The standard Cobb-Douglas model allows for the former, and the notion of constant elasticity of substitution between capital and labor is embodied in the so-called CES production function introduced by Arrow et al. (1961). An alternative possibility is to rule out the possibility of capital deepening without accompanying technological change, that is, assuming that it is impossible to incorporate technological improvements without embedding them in new capital equipment (for example, Solow et al. 1966). However, while the two phenomena - capital deepening versus technological advance - can be distinguished in principle, there is apparently no satisfactory test to distinguish them in practice (Nelson 1973). Evidently, in the real world, virtually all opportunities for substitution require time and technological investment. The greater the degree of substitution, the more time and investment may be needed. We postulate that movements of the frontier are reflected and can be captured in time series data over a long enough period.

The need to distinguish between short-term and longer-term behavior seems to have been noticed in a different context by Levine (1960) and Massell (1962). It was rediscovered by Nelson (1973). The problem is that the sum of incremental short-term changes in the contributions of the factors of production (K, L) do not necessarily account for long-term changes. In Nelson's words (ibid., p. 465):

Experienced growth is unlikely to be the simple sum of the contributions of separate factors. One could take the position that the degree of interaction among the factors is small, and that the separable contributions of the different factors are like the first terms of a Taylor expansion. This is an arguable position, but it rests on an assumption about the nature of the production function and about technical change. The approximation might be good and it might be poor. If the time period in question is considerable, Taylor series arguments are questionable.

Since the Cobb-Douglas and CES functions do not exhibit sharply changing gradients, it seems likely that interaction terms will have to be incorporated in the production function.

There is a further difficulty, namely that the three driving variables -and especially capital and useful work - are also to some extent complements. Machines need workers to operate and maintain them, and they need energy to function. In other words, they must be present in fixed (or nearly fixed) combinations. There is ample statistical, as well as anecdotal, evidence of complementarity between energy and capital (for example, Berndt and Wood 1975). This situation is inconsistent with the Cobb-Douglas production function or, indeed, any other smooth function of two or three variables. A production function with fixed ratios of inputs is called a Leontief function, because fixed ratios of inputs are characteristic of the Leontief model. Note that the plot of a Leontief production function in two (or three) dimensions is like a right angle or a corner. Except at the point of intersection (the corner), either some capital, or some labor (or some X) will be unutilized. It is not a smooth or differentiable function.

Assuming that aggregate production functions can be justified at all, the real situation at the national level is certainly somewhere in between the Cobb-Douglas and Leontief cases. That is to say, a realistic production function allowing for some degree of complementarity as well as some substitutability may not incorporate a sharp corner, but it should exhibit a sharply changing gradient, in the range where substitution is possible, as well as with a maximum second derivative near the optimum combination of the three variables. The three cases are shown graphically in Figure 6.1.

Another major problem is estimating capital stock per se. As we noted in the previous chapter, the so-called Cambridge controversies in the 1960s highlighted many of the problems, notably the difficulty of aggregating heterogeneous capital-comprising machines, structures, inventories, infrastructures, money and even natural resource stocks (Harcourt 1972). In practice, we adopt Maddison's 'perpetual inventory' method (PIM) to measure capital in monetary terms, accumulating capital from new investment less depreciation (Maddison 1982). But this method has certain drawbacks. As a subtraction from potential consumption, it makes reasonable sense, but it makes no allowance for changes in monetary values arising from price fluctuations, or for the non-equivalence and non-substitutability of different kinds of capital within the category. Machines are not equivalent to or interchangeable with structures or inventories, and a truck is not equivalent to 100 wheelbarrows. Indeed, some other implicit assumptions of neoclassical production theory can be violated. Unfortunately, no one knows how seriously these distortions bias the results.

The next class of difficulties concerns estimation of the parameters of the production function by regressing time-series data for a few highly correlated variables (for example, Mendershausen 1938; Griliches and Mairesse 1998). It was discovered long ago that almost any set of collinear capital and labor time series can be fitted to a Solow-type Cobb-Douglas function with a residual A(t) subject to the Euler condition (constant returns) and constant savings rate. This is partly due to the fact that the residual A(t) absorbs deviations from the actual data (for example, Hogan 1958). For other critiques along these lines see Shaikh (1974), Simon (1979) and Shaikh (1980).

More recently the problem with production functions has been restated more broadly by Felipe and Fisher as follows:

The ex post income accounting identity that relates the value of output (VA) to the sum of the wage bill (wL where w is the average wage rate and L is employment) plus total profits (rK where r is the average ex post profit rate and K is the stock of capital) can be easily rewritten through a simple algebraic transformation as VA = A(t)F(K, L) . . . The implication of this argument is that the precise form . . . corresponding to the particular data set VA = wL + rK has to yield a perfect fit if estimated econometrically (because all that is being estimated is an identity); the putative elasticities have to coincide with the factor shares and the marginal products have to coincide with the factor prices . . . it says nothing about the nature of production, returns to scale and distribution. (Felipe and Fisher 2003, pp. 252-3)

Felipe and Fisher also note that the accounting identity does not follow from Euler's theorem if the aggregate production function does not exist. Finally, the ex post profit rate r in this identity is not the same as the cost of capital to users; it is merely the number that makes the accounting identity hold (ibid).

A consequence of this is that a production function derived from empirical data cannot be used to determine output elasticities with high reliability. Apart from the implicit accounting identity, estimated parameters tend to pick up biases from mis-specification or omitted variables. For us, a further question is whether the third variable in our formulation (exergy or useful work) really captures enough of the impact of other aspects of technological advancement, structural change and human capital. We will attempt to address this question again later.

In some ways, the case against using aggregate production functions of a very few variables seems overwhelming; certainly stronger than the case for using them.2 The major reason for taking this approach, despite problems, is that it is familiar and both relatively transparent and relatively convenient. The conclusions, if any, must, necessarily, be considered carefully in the light of the criticisms.

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