How Sound are the Foundations of the Aggregate Production Function

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Abstract:   The aggregate production function has been subject to a number of criticisms ever since its first empirical estimation by Cobb and Douglas in the 1920s, notably the problems raised by aggregation and the Cambridge Capital Theory Controversies. There is a further criticism due initially to Phelps Brown (and elaborated, in particular, by Simon and Shaikh) which is not so widely known. This critique is that because at the aggregate level only value data can be used to estimate production function, this means that the estimated parameters of the production function are merely capturing an underlying accounting identity. Hence, no reliance can be placed on estimates of, for example, the elasticity of substitution as reflecting technological parameters. The argument also explains why good statistical fits of the aggregate production functions are obtained, notwithstanding the difficulties posed by the aggregation problem and the Cambridge Capital Controversies noted above. This paper outlines and assesses the Phelps Brown critique and its extensions. In particular, it considers some possible objections to his argument and demonstrates that they are not significant. It is concluded that the theoretical basis of the aggregate production function is problematic.

Keywords: Cobb-Douglas, production function, income identity

It is somewhat paradoxical that one of the concepts most widely used in macroeconomics, namely the aggregate production function, is the one whose theoretical rationale is perhaps most suspect. The serious problems raised by the Cambridge Capital Theory Controversies dominated ¡§high theory¡¨ in the late 1960s and early 1970s, and eventually led to an agreement that reswitching and capital reversing were theoretically possible (Harcourt, 1972). This posed serious problems for the justification of the use of the neoclassical one-sector aggregate production function as a ¡§parable¡¨. The revival of interest in growth theory with the development of endogenous growth theory is still squarely in the tradition of the neoclassical growth model. Pasinetti (1994) was compelled to remind the participants at a recent conference on economic growth that

this result [that there is no unambiguous relationship between the rate of profit and the capital-labour ratio], however uncomfortable it may be for orthodox theory, still stands. Surprisingly, it is not mentioned. In almost all ¡¥new growth theory¡¦ models, a neoclassical production function, which by itself implies a monotonic inverse relationship between the rate of profits and quantity of capital per man, is simply assumed. (emphasis in the original)

If this were not enough, there is a whole series of further problems concerned with the question of whether or not micro-production functions can be aggregated to give a macro-relationship which can be shown to reflect the underlying technology of the economy in some meaningful way (Walters, 1963, Fisher, 1987, 1992). Indeed, Blaug (1974), who can be scarcely viewed as sympathetic to the Cambridge UK view of the interpretation, or importance, of the Capital Theory Controversies, nevertheless considers that the aggregation problem effectively destroys the rationale of the aggregate production function. ¡§Even if capital were physically homogeneous, aggregation of labour and indeed aggregation of output would still require stringent and patently unrealistic conditions at the economy¡Vlevel¡¨.  Moreover, ¡§the concept of the economic meaningful aggregate production function requires much stronger and much less plausible conditions than the concept of an aggregate consumption function. And yet, undisturbed by Walter¡¦s conclusions or Fisher¡¦s findings, economists have gone on happily in increasing numbers estimating aggregate production functions of even more complexity, barely halting to justify their procedures or to explain the economic significance of their results.¡¨ (Blaug, 1974).

The defence of this procedure, however, has been eloquently put forward by Solow (1966), who can hardly be accused of not being fully aware of the aggregation problem. ¡§I have never thought of the macroeconomic production function as a rigorously justifiable concept. ¡K It is either an illuminating parable, or else a mere device for handling data, to be used so long as it gives good empirical results, and to be abandoned as soon as it doesn¡¦t, or as soon as something else better comes along¡¨ (Solow, 1966). Wan (1971) reflects this view when he argues that Solow¡¦s (1957) path-breaking approach may be defended on the grounds that ¡§one may argue that the functional relation between Q and K, L is an ¡¥empirical law¡¦ in its own right. In the methodological parlance of Samuelson, this is an operationally meaningful law, since it can be empirically refuted¡¨. (Q, K and L are output, capital, and labour.) From the first studies by Douglas and his various collaborators (see, for example, Cobb and Douglas, 1928, Douglas, 1944 and 1976) it has often been found that the aggregate production function gives a close statistical fit, especially using cross-sectional data, with the estimated output elasticities close to the factor shares.

However, the problem is that these studies generally could not have failed to find a close correspondence between the output elasticities and the factor shares. This arises from the fact that ideally the production function is a microeconomic concept, specifying the relationship between physical outputs and inputs (such as numbers of widgets, persons employed, and identical machines). On the other hand, at the aggregate level, constant price value data are used for capital and output.  Yet this is not an innocuous procedure, as it has been argued that this undermines the possibility of empirically testing the aggregate production function.  As Simon (1979a) pointed out in his Nobel Memorial lecture, the good fits to the Cobb-Douglas production function ¡§cannot be taken as strong evidence for the classical theory, for the identical results can readily be produced by mistakenly fitting a Cobb-Douglas function to data that were in fact generated by a linear accounting identity (value of output equals labor cost plus capital cost)¡¨.

Specifications of aggregate production functions, using value data, may be nothing more than approximations to an accounting identity, and hence can convey no information, per se, about the underlying technology of the ¡§representative firm¡¨. This is not a new critique, but first came to prominence in Phelps Brown¡¦s (1957) criticism (later formalised by Simon and Levy (1963)) of Douglas¡¦s cross-industry results.  Shaikh (1974, 1980, 1987) generalised it to time-series estimation of production functions and Simon (1979b) also considered the criticism in the context of both cross-section and time-series data. The criticism was re-examined and extended by Felipe and McCombie (2000, 2001 a&b, 2002 a&b), Felipe (2001a and 2001b), McCombie (1987, 1998, 2000, 2000-2001, 2001), McCombie and Dixon (1991) and McCombie and Thirlwall (1994).

Once it is recognised that all that is being estimated is an underlying identity, it can be shown how it is always possible, with a little ingenuity, to obtain a perfect statistical fit to a putative production function, which exhibits constant returns to scale and where the estimated ¡§output elasticities¡¨ equal the factor shares. It can also shown how the results of estimation of production functions which find increasing returns to scale and externalities are simply due to misspecification of the underlying identity and the estimated biased coefficients may actually be predicted in advance (McCombie, 2000-2001, Felipe, 2001a).

The purpose of this paper is to provide a survey of the main elements of this critique and also to consider some counter-criticisms which have been made.  We conclude that the latter leave the central tenet unaffected.

The Cobb-Douglas Production Function and the Accounting Identity

The problem that the accounting identity poses for the interpretation of the aggregate production function was first brought to the fore by Phelps Brown (1957) in his seminal paper ¡§The Meaning of the Fitted Cobb-Douglas Function¡¨.  (It had also been partly anticipated by Bronfenbrenner (1944).)  It is one of the ironies of the history of economic thought that this article, which challenged the whole rationale for estimating aggregate production functions, was published in the same year as Solow¡¦s (1957) ¡§Technical Change and the Aggregate Production Function¡¨. The latter, of course, was largely responsible for the beginning of the neoclassical approach to the empirical analysis of growth.

Phelps Brown¡¦s critique was addressed to the fitting of production functions using cross-sectional data and was specifically directed at Douglas¡¦s various studies (see Douglas, 1944), and we consider this first, before considering the case of time-series data.

The fact that the crucial tenet of Phelps Brown¡¦s (1957) argument was presented rather obscurely and was buried in his paper did not help its reception, even though it was published in one of the US¡¦s leading economics journal. The following is the key passage:

The same assumption would account for the observed agreement for the values obtained for Ą [the output elasticity of labour], and the share of earnings given by the income statistics. For on this assumption the net products to which the Cobb-Douglas is fitted would be made up of just the same rates of return to productive factors, and quantities of those factors, as also make up the income statistics; and when we calculate Ą by fitting the Cobb-Douglas function we are bound to arrive at the same value when we reckon up total earnings and compare them with the total net product. In Ą we have a measure of the percentage change in net product that goes with a 1 per cent change in the intake of labour, when the intake of capital is held constant; but when we try to trace such changes by comparing one industry with another, and the net products of the two industries approximately satisfy, Vi = wLi + rJi, the difference between them will always approximate to the compensation at the wage rate w of the difference in labour intake. The Cobb-Douglas Ą and the share of earnings in income will be only two sides of the same penny. (Phelps Brown, 1957, p.557)

V, w, L, r, and J are output (value added at constant prices), the average wage rate, the labour input, the average observed rate of return and the constant price value of the capital stock.   (We use V and J to refer to the value measures; Q and K are used below to denote the physical measures of output and capital.) The subscript i denotes the ith firm or industry.

The argument therefore seems to be this. The output elasticity of the Cobb-Douglas production function is defined as ƒÑ = („gV/„gL)(L/V) and given the assumptions of the neoclassical theory of factor pricing, the marginal prodiuct of labour equals the wage rate, „gV/„gL = w. Given these assumptions, it also follows that ƒÑ = a =  wL/V, the share of labour in value added.

However,  as Phelps Brown pointed out in the quotation cited above, there is also an accounting identity that defines the measure of value added for all units of observation, whether they be the firm, the 1, 2, 3 or 4 digit SIC, or the whole economy. This is given by:

It should be emphasised that there are no behavioural assumptions underlying this equation, in that it is compatible with any degree of competition, increasing or decereasing returns to scale and the existence or not of a well-baehaved underlying production function. J is the constant price measure of the value of the capital stock (normally calculated by the perpetual inventory method) and r is the observed rate of profit normally calculated as the product of the share of profits in value added (1-Ą) and the output-capital ratio, i.e., r = (1-Ą)V/K

Consequently, partially differentiating the accounting identity, Vi = wLi + rJi, with respect to L gives „gV/„gL = w and  it follows that („gV/„gL)/(L/V) = a = wL/V. The argument stemming from the identity has not made any economic assumptions at all (e.g., it does not rely on the marginal productivity theory of factor pricing or the existence of perfectly competitive markets and optimising behaviour of firms). Consequently, the finding that the putative output elasticities equal the observed factor shares cannot be taken as a test that factors of production are paid their marginal products. This is a position, however, that was not accepted by Douglas (1976) himself. ¡§A considerable body of independent work tends to corroborate the original Cobb-Douglas formula, but more important, the approximate coincidence of the estimated coefficients with the actual shares received also strengthens the competitive theory of distribution and disproves the Marxian.” However, it is noticeable that, in his survey, Douglas fails to mention the Phelps Brown (1957) paper.

If the output elasticity of labour and the share of labour¡¦s total compensation are merely ¡§two sides of the same penny¡¨, could it be that the Cobb-Douglas is simply an alternative way of expressing the income identity and, as such, has no implications for the underlying technology of the economy? This was the proposition that Simon and Levy (1963) proved some eight years later.

Following Simon and Levy (1963) and Intriligator (1978), the isomorphism between the Cobb-Douglas production function and the underlying accounting may be simply shown. The Cobb-Douglas, when estimated using cross-section (firm, industry or regional) data, is specified as:

where  and   are the values of some reference observations, such as those of the average firm or the base year.

The following approximation holds for any variable X, when X and   do not greatly differ:

Consequently, equation (2) may be written as:

A comparison with the income identity, namely,  Vi „k wLi + rJi ,        shows that w = ƒÑ  or w  = a = ƒÑ. A similar relationship holds between the output elasticity of capital and capital¡¦s share. Moreover, (1-ƒÑ-ƒÒ) equals zero, so that the data will always suggest the existence of constant returns to scale, whatever the true technological relationship.

What is the implication of all this?  Start from the accounting identity and undertake the approximation in the reverse order from that outlined above. The two procedures are formally equivalent. The consequence of this argument is that, for reasonably small variations of L and J and with w and r constant (the last two are not essential, as we shall see below), a Cobb-Douglas multiplicative power function will give a very good approximation to a linear function. Since the linear income identity exists for any underlying technology, we cannot be sure that all that the estimates are picking up is not simply the identity. The fact that a good fit to the Cobb-Douglas relationship is found implies nothing, per se, about such technological parameters as the elasticity of substitution.

To see this consider Figure 1, which shows the accounting identity expressed as Vi/Li= w + rJi/Li and the Cobb-Douglas relationship as Vi/Li = A(Ji/Li)a. The observations must lie exactly on the accounting identity.  We have assumed, for the moment, that w and r are constant. (If they show some variation, then the observations would be a scatter of points around the line where the slope and the intercept represent some average value of r and w.) The Cobb-Douglas approximation is given by the solid curved line,  cd,  which is tangent to the  income identity, ab. If, however, we mistakenly statistically fit a Cobb-Douglas function to these data, we will find the best fit depicted by the dotted curved line, ef. The residuals will be autocorrelated.

Of course, the empirical question arises as to how much variation in the data is required before the Cobb-Douglas ceases to give a plausible fit to the data. Fortunately, Simon (1979b) has provided the answer. He calculated the ratio between the predicted value given by the Cobb-Douglas function (VCD) and that by the accounting identity, VA, namely, VCD/VA = A(L/J)ƒÑ/(r + w(L/J)). He found that when the L/J ranged from 16 to 1, the greatest error, or ratio, was only 10 per cent. He concluded ¡§since in the data actually observed, most of the sample points lie relatively close to the mean value of L/J, we can expect average estimating errors of less than 5 per cent.¡¨ (Simon 1979b).

The good approximation of the Cobb-Douglas to the accounting identity is also likely to carry through even when we allow w and r to change, provided the factor shares do not show very much variation. To see this, assume a continuum of firms and differentiated the accounting identity to give:

dVi = (dwi)Li+ widLi + (dri)Ji+ ri dJi                                   (7)

dVi/Vi = aidwi/wi + aidLi/Li + (1-ai)dri/ri + (1-ai) dJi/Ji    (8)

Let us assume factor shares are constant (and there are many reasons why this should occur other than because there is a Cobb-Douglas production function, e.g. firms pursue a constant mark-up pricing policy). Equation (8) may be integrated to give:

where B is the constant of integration.

Provided that wiari(1-a) shows very little variation or is orthogonal to

LiaJi(1-a)  or both, the putative Cobb-Douglas production function will once again give a very good fit to the data.

It should be noted that this argument is not just confined to the Cobb-Douglas production function. Simon (1979b) explicitly considers the CES production function given by V = ƒ×(ƒÔL-ƒâ + (1-ƒÔ)J-ƒâ)-(1/ƒâ), where ƒ×, ƒâ, and ƒÔ   are parameters. He argues that if the true relationship were given by the accounting identity and we were mistakenly to estimate the CES production function, then if ƒâ goes to zero, the function becomes a Cobb-Douglas. He cites Jorgenson (1974) as suggesting that most estimates give ƒâ  close to zero and so the argument still applies. However, more recent studies find that the putative aggregate elasticity of substitution is less than unity. But the argument is more general than Simon implies. If we were to express any production function of the form Vi =f(Li, Ji) in proportionate rates of change, we would find that dVi/Vi = c +ƒÑidLi/Li + ƒÒi dJi/Ji which is formally exactly equivalent to the accounting identity, provided aidwi/wi + (1-ai)dri/r is again roughly constant or orthogonal to aidLi/Li + (1-ai) dJi/Ji.  This may be seen from equation (8) from which it also follows that ƒÒi = (1-ƒÑi) = (1-ai). If shares do vary, then we may be able find an explicit functional form that is more flexible than the Cobb-Douglas (such as the CES) that gives a good fit to the accounting identity; but, of course, this does not mean that the estimated coefficients can now be interpreted as technological parameters.  If aidwi/wi + (1-ai)dri/r does not meet the assumptions noted above, all this means is that the estimate functional form will be misspecified and the goodness of fit will be reduced (McCombie, 2000 and Felipe and McCombie, 2001).

The fact that the identity precludes interpreting the Cobb-Douglas or more flexible functional forms as unambiguously reflecting the underlying technology of the economy implicitly suggests that this is true of estimations using time-series data.      Nevertheless, the arguments, as Shaikh (1974, 1980) has shown, follow through in the case of time-series data. Differentiating the income identity with respect to time, we obtain

vt = at t + (1-a)t t + atƒÜt + (1-a)tjt   (10)

where v, ƒÜ, j,  , and  denote exponential growth rates.  Assuming that factor shares are constant and integrating, we obtain:

Let us assume that the growth of wages occurs at a roughly constant rate and the rate of profit shows no secular growth (both of which may be regarded as stylised facts). Consequently, at t + (1-a)t t „l ƒÜ, a constant, and so equation (11) becomes the familiar Cobb-Douglas with exogenous technical change, namely Vt = AoeƒÜtLtaJt(1-a).

In fact, while the cross-section studies normally give a very good fit to the Cobb-Douglas (and other) production functions, the time-series estimations sometimes produce implausible estimates with, for example, the coefficient of capital being negative. The fact that the results are often so poor may ironically give the impression that the estimated equation is actually a behavioural equation. However, the failure to get a good fit will occur if either the factor shares are not sufficiently constant or the approximation at t + (1-a)t t „l ƒÜ  is not sufficiently accurate. In practice, the latter proves to be the case, as estimations of equation (11) with a variety of data sets produces well-determined estimates of the coefficients with low standard errors (notwithstanding the ever-present problem of multicollinearity). It transpires that the rate of profit has a pronounced cyclical component and so proxying the sum of the weighted logarithms of w and r  by a linear time trend (or their growth rates by a constant) biases the estimated coefficients of lnL and lnJ (McCombie, 2000-2001, Felipe and Holz, 2001, and Felipe and McCombie, 2001b).

The conventional neoclassical approach, which is based on the maintained hypothesis that an aggregate production function is, in fact, being estimated, usually attributes a poor fit to the failure to adjust the growth of factor inputs for the changes in capacity utilisation. Since ƒÜt „k at t + (1-a)t t   tends to vary procyclically, the inclusion of a capacity utilisation variable (or the adjustment of k and ƒÜ for changes in their utilisation rates) will tend to improve the goodness of fit and cause the estimated coefficients to approximate more closely the relevant factor shares. As Lucas (1970), commented: ¡§…some investigators have obtained ¡¥improved¡¦ empirical production functions (that is, have obtained labor elasticities closer to labor¡¦s share) by ¡¥correcting¡¦ measured capital stock for variations in utilisation rates¡¨ (Lucas, 1970). An alternative procedure would be to introduce a sufficiently complex non-linear time trend more accurately to capture the variation of ƒÜt (Shaikh, 1980, Felipe and McCombie, 2001c). With sufficient ingenuity, we should be able eventually to approximate closely the underlying identity, increasing both the R2 and the values of the t-statistics, and hence find a very good fit for the ¡§production function¡¨. Generally, as we have noted above, it is this problem, rather than the change in factor shares, that is of greater empirical importance.

The Problems of Using Monetary Values at Constant Prices as Proxies for Quantities

The problem ultimately stems from the use of value data as a proxy for ¡§quantity¡¨ or ¡§volume¡¨ measures. The fact that the neoclassical production function should be theoretically specified in terms of physical quantities, and not in value measures, is not often explicitly stated, but an exception is Ferguson (1971).  In his comment on Joan Robinson¡¦s review of his book, The Neoclassical Theory of Production, he argues, ¡§I assume a production function relating physical output to the physical inputs of heterogeneous labour, heterogeneous machines and heterogeneous raw materials. As a first approximation, I further assume that the definition of the output required the various raw materials to be used in fixed proportions. Thus, attention was directed to the first two heterogeneous categories of inputs. Assuming variable proportions, each physical input has a well-defined marginal physical product. If profit maximisation is also assumed  … each entrepreneur will hire units of each physical input until the value of its marginal physical product is equal to its market determined and parametrically given input price¡¨ (emphasis in the original). He continues that ¡§neoclassical theory deals with macroeconomic aggregates, usually by constructing the aggregate theory by analogy with the corresponding microeconomic concepts. Whether or not this is a useful concept is an empirical question to which I believe an empirical answer can be given. This is the ¡¥faith¡¦ I have but which is not shared by Mrs Robinson. Perhaps it would be better to say that the aggregate analogies provide working hypotheses for econometricians.¡¨ (Ferguson, 1971.)  Thus, Ferguson is implicitly arguing that although the neoclassical production function should be specified in terms of physical quantities, these may be adequately proxied by deflated monetary values where this is necessary for aggregation. Jorgenson and Griliches (1967), for example, take a similar position, explicitly stating that physical output is taken as ¡§real product as measured for the purposes of social accounting¡¨.

If we are dealing with physical quanitities, then it is possible to estimate a production function and to test the marginal producivity theory of distribution. To see this, consider the neoclassical approach that uses a micro-production function specified in physical terms (assuming no technical change):

where Q and K are measured in physical or homogeneous units.

The first order conditions are, under the usual assumptions, „gQ/„gL = fL and „gQ/„gK = fK, where  fL  is the marginal product of labour (measured in physical units of output, say, widgets per worker) and fK  is the marginal product of capital measured in terms of widgets per ¡§leet¡¨ (where the capital stock, after Joan Robinson, is measured as the number of leets).

Let us assume that the production function is homogeneous of degree one. By Euler¡¦s theorem which, of course, has no economic content, per se, we have:

It should be noted that equation (13) follows from equation (12) purely as a mathematical proposition. It is given economic content by assuming that factors are paid their marginal products. If we wish to express equation (13) in monetary terms, it is multiplied by the price of widgets so that:

where m and n are the marginal products of labour and capital in monetary terms, i.e., m = pfL and n = pfK.    It should be emphasised that the physical quantities can always be simply recovered from the data by dividing by the price. Let us, for expositional purposes, assume that the marginal products are (roughly) constant and that equation (12) is a Cobb-Douglas production function.  If Qt = ALt ƒÑKt (1-ƒÑ)  were to be estimated as lnQt = c + b1lnLt + b2lnKt, the estimates of b1 and b2  would be the relevant output elasticities. The accounting identity does not pose a problem in this case. If we were to estimate the linear equation (14) as:

we would find the estimates of b3 and b4 would be pfL  and pfK , or m and n.  If the output elasticities of labour and capital are 0.75 and 0.25, then m/p = 0.75Q/L and n/p = 0.25Q/L.

Does the identity pose a problem for the interpretation of the production function? The answer is clearly no. Let us assume that factors are not paid their marginal products. We are dealing, say, with a command economy and labour only receives ƒ× of its marginal product (where 0>ƒ× >1) while the state, the owner of capital, appropriates the remainder, plus the payments accruing to capital. Thus, we have a distribution equation pQ = wL + rK where w = ƒ×m = ƒ×pfL and r = (1-ƒ×)pfK.  There is of course an infinite number of combinations of w and r that could satisfy equation (15), but if we were to estimate it, then the coefficients would be m and n. Equation (14) can be interpreted as just a linear transformation of equation (12).  This is an important point, because the coefficients of the estimated linear equation (15) are determined by the underlying production function and they will differ from the observed wages and rate of profit, if factors are not paid their marginal products. This is because Q is a physical measure and is independent of the distribution of the product.

Thus, in these circumstances, the discussions concerning the appropriate estimation procedures of the production function (whether it should be part of a simultaneous equation framework, etc.) become relevant.  Moreover, the marginal productivity theory of factor pricing may be tested by a comparison of the estimated output elasticities with the factor shares.

Alternatively, it may be assumed that factors are paid their marginal products and the growth accounting approach may be adopted. There is a well-defined neoclassical production function Q = f(L,K,t) which, when expressed in growth rates, becomes qt = ƒÜt + ƒÑt ƒÜt + (1-ƒÑt)kt  where ƒÑt = (fLL/Q) and (1-ƒÑt) = (fKK/Q). Since factors are paid their marginal products, pfL = w and pfk = r and so the Solow residual (or the growth of total factor productivity) is given by

ƒÜt „k qt – atƒÜt -(1-at)kt . Of course, we do not need to estimate a production function to obtain the Solow residual, but the whole procedure depends upon the existence of perfect competition and the marginal productivity theory of factor pricing and so ultimately requires that these assumptions are capable of being tested empirically. Alternatively, we may calculate the Solow residual from the ¡§dual¡¨as ƒÜt „k at +(1-a)t . But it must be emphasised that all the usual neoclassical assumptions underlie this interpretation, including that there is a well-behaved production function and factors are paid their marginal products.

The neoclassical approach then moves almost seemlessly from the consideration of the production function in terms of physical quantities to the use of value data, where it assumed (erroneously) that all the arguments follow through in a straightforward manner. The proposition that the aggregate production function should be regarded simply as a ¡§parable¡¨ crops up time and time again in the defence of the neoclassical approach, especially with regard to the Cambridge Capital Theory debates. The world is a complex place, so the argument goes, and any model necessarily abstracts from reality. It may be, for example, that the strict conditions for the aggregation of production functions are not met theoretically. But if the estimation of an aggregate production function gives a good statistical fit and plausible estimates of, for example, the output elasticities, we can have Ferguson¡¦s faith that the estimated relationship is telling us something about the underlying technology of the economy.

This is very much reminiscent of Friedman¡¦s (1953) instrumentalist approach to methodology. The realism or otherwise of the assumptions is irrelevant ¡V what matters is the predictive power of the model.  Of course, it could be argued that considering whether or not the estimates are plausible rather begs the question. But we may absolve the argument from this charge at least, as we do have some indication of what is considered plausible. Fisher (1971) has noted that Solow has commented that if Douglas had found capital¡¦s share to have been three-quarters rather than one-quarter, we should not now be talking about production functions. But the problem that we encounter is that in moving from the micro- to the macro-level we need prices to aggregate the output.

The difficulty is that with the use of value data, the underlying accounting identity does produce an insurmountable problem. Let us assume that firms pursue a mark-up pricing policy where the price is determined by a fixed mark-up on unit labour costs. (We make this assumption for simplicity as, in practice, firms mark-up on normal unit costs. See Lee (1999) for a detailed discussion.) Thus pi = (1+ƒài)wiLi/Qi where ƒà  is the mark-up.   The value added is Vi = piQi = (1+ƒài)wiLi  and for industry as a whole V = „¸ piQi  = „¸(1+ƒài)wiLi, or   approximately, V = (1+ƒà)wL, where ƒà is the average mark-up and w is the average wage rate. Labour¡¦s share is a =1/(1+ ƒà) and will be constant to the extent that the mark-up does not vary. In practice, it is likely to vary to the extent that the composition of firms with differing mark-ups alter and there are changes in the individual mark-ups, which may be temporary, as a result of the wage bargaining process. We also have the identity V „k wL + rJ where rJ, the operating surplus, is equal to ƒàwL and (1-a) = ƒà/(1+ƒà). The identity now poses a major problem for the aggregate production function. Suppose that w and r do not change over time. If we were to estimate  V = b6L + b7J, then the estimates of the b6 and b7 will always be w and r respectively. If factor shares are constant, then an approximation to the accounting identity will be given by V = ALaJ(1-a), but the causation is from the identity to the multiplicative power function, not the other way around. The values of the putative elasticities are determined by the value of the mark-up and do not reflect any technological relationship. The fact that the Cobb-Douglas gives a good fit to the data does not imply that the aggregate elasticity of substitution is unity. Shares may vary, in which case a more flexible function than the Cobb-Douglas, such as the translog, will give better fit; but we still cannot be sure that the data is telling us anything about the underlying technology of the economy.

It is worth elaborating on the value added  accounting identity. The estimate of r in the accounting identity, V „k wL +rJ,  is the observed aggregate rate of profit or rate of return. It has been used, for example, in numerous studies concerned with the analysing the profit squeeze that occurred in the advanced countries during part of the post-war period. However, the impression may be given that somehow the critique depends upon this method of calculating r explicitly from the accounting identity.  It clearly does not. Value added must, if it is to be accurately measured, be the sum of the total compensation of labour and capital (and land, but for expositional ease we shall ignore this). Since we have direct statistics on wages and employment, the rate of return is imputed from the data on total profits and the gross or net capital stock, the latter being calculated by the perpetual inventory method. However, the estimate of value added must equal the compensation of employees and the self-employed and the gross (or net) operating surplus. If we did have an independent measure of the rate of return that differed from the imputed value, then the statistical discrepancy would have to be resolved. ,   The rate of profit calculated from the identity and the national accounts will contain any elements of monopoly profits that accrue to the firms, as will the estimate of value added. Hill (1979) contains a detailed discussion of the various types of profit measures, and identifies the conditions in which the net and gross rates of return will approximate to average realised internal rates of return.

A misunderstanding may arise because it is sometimes assumed that a firm¡¦s cost identity is given by C = wL + rcK where C is the total cost and rc is the competitive rate of return or the competitive cost of capital, such that economic profits are zero. It is also usually assumed that the labour market is competitive so w is the competitive wage. rc is sometimes calculated as the rental price of capital.   In this case, the rental price of capital will only equal the rate of return if the former has been calculated correctly and perfect competition prevails. If it is assumed that markets are competitive, then  ¡§given either an appropriate measure of the flow of capital services or a measure of its price, the other measure may be obtained from the value of income from capital¡¨ (Jorgensen and Griliches, 1967) —  although the rider should perhaps be added that the income from capital (and hence the measure of value added) should exclude the monopoly profits. However, because the conditions for producer equilibrium have been invoked, Jorgenson and Griliches continue that ¡§the resulting quantity of capital may not be employed to test the marginal productivity theory of distribution, as Mrs Robinson and others have pointed out¡¨. Of course, this does not prevent the neoclassical economist from estimating an aggregate production function, but the ¡§quantity¡¨ of output should, as we have noted, in these circumstances, be the constant price value of total costs.  However, even if the estimate of rc is subject to serious measurement errors, if this procedure is followed we should still find that that the output elasticities equal the factor shares so long as the total cost identity holds by construct. It should be noted, however, that the national accounts data for value added are not constructed on the basis of any of these neoclassical assumptions, but from actual magnitudes. Hence while estimating production functions is a tautology, it is not a tautology generated by constructing the data under neoclassical assumptions.

Micro-production Functions and the Aggregate Cobb-Douglas ¡§Production Function¡¨

As we have seen, so long as factor shares are constant, we will obtain a good statistical fit to the supposed Cobb-Douglas production function even though it does not represent the underlying technology. There have been a number of studies that have explicitly illustrated how this can occur under a variety of assumptions.

Houthakker (1955-6) developed a model where, although each industry was subject to a fixed coefficients technology, the aggregate data behaved as if it were a Cobb-Douglas production function. (A simple explanation of this model may be found in Heathfield and Wibe (1987, pp.150-152).) He assumes that there are machines in existence that can produce output for every fixed pair of input coefficients and that the input-output ratios are distributed in ascending order according to a Pareto distribution. He shows that these micro-production functions can be aggregated into an industry-wide Cobb-Douglas production function with decreasing returns to scale. This result clearly shows how an aggregate production function may give the appearance of a technology that has an elasticity of substitution of, in this case, unity, whereas at the micro level there is no possibility of smoothly substituting between inputs. The implications of this have been succinctly stated by Blaug (1974). ¡§It is well known that the competitive theory of factor pricing does not stand or fall on the existence of continuous and differential production functions: we can handle fixed coefficients simply by writing our equilibrium conditions as inequalities rather than equalities. True, but the ability to fit an aggregate production of the Cobb-Douglas form may throw no light on the underlying technology of the firm and hence on the process by which competitive pressures in individual markets impute prices to factors of production, and that is the question at issue.¡¨ Of course, it may be argued that the fact that the firm sizes have to be distributed in a particular way, which is unlikely to occur in reality, makes this merely a curiosity. Nevertheless, it does serve to demonstrate theoretically how a good fit to an aggregate production function may give a completely misleading picture of the underlying technology of the economy.

Fisher (1971) has likewise raised serious doubts concerning the aggreg.............

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