Copyright © The Authors 2007. Published by Oxford University Press.
The long-term sucCESs of the neoclassical growth model
* Johann Wolfgang Goethe University, e-mail: klump{at}wiwi.uni-frankfurt.de
** European Central Bank, e-mail: peter.mcadam{at}ecb.int
*** European Central Bank, e-mail: alpo.willman{at}ecb.int
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In this paper, we seek to re-establish the link between the constant elasticity of substitution (CES) production function and neoclassical Solow growth theory. We do so in three dimensions. First, we review the increasing importance of the CES technology in modern dynamic macroeconomics, in expanding not only theory but also in addressing important policy questions. Second, we aue that the importance of the CES function in growth theory is intimately linked to ‘normalization’. Finally, we examine the data congruence between CES functions and recent growth patterns in the USA and the euro-area economies, where we apply a supply-side system incorporating a CES function with factor-augmenting and time-varying technical progress.
Key Words: Solow growth model • capital–labour substitution • technological change • factor shares • normalized CES function • supply-side system • United States • euro area
We are grateful to our discussant Bob Solow, as well as to Chris Allsopp, Bob Chirinko, and seminar participants at the American Economic Association session, ‘The 1956 Contribution to Economic Growth Theory by Robert Solow’, Boston, MA, January 2006, for helpful comments and discussions. The opinions expressed here are not necessarily those of the European Central Bank. McAdam is also Honorary Reader at the University of Kent.
1 Indeed, analysing growth theory outcomes conditional on different functional forms for aggregate production was a significant, if then at least arguably neglected, part of the Solow (1956) paper; on pages 73–8, he analyses Leontief, Cobb–Douglas, and CES production functions as solutions of the famous ‘fundamental equation’ of the model.
2 As Chirinko and Mallick (2005) have recently termed it, since 1956 the ‘Solow interval’ (as a range of elasticities of substitution compatible with a stable steady state) has replaced the ‘Harrod–Domar knife-edge’.
3 See Jones (2003) for a stout defence of Cobb–Douglas technology in aggregate growth models.
4 The estimation of the single production function, however, can only be accomplished with quite restrictive assumptions about the nature of technological progress. Antràs (2004), for instance, showed that the assumption of Hicks-neutral technical progress, popular in studies of the elasticity of substitution, together with the observed development of factor-income shares can lead to significant omitted-variable biases. As Antràs demonstrated, with the a priori assumption of a common growth rate for labour- and capital-augmenting technical change, a relatively stable factor share, and a rising capital intensity in the long run, it is a logical conclusion that Berndt (1976) found a Cobb–Douglas function with an elasticity of substitution equal to one that should fit best for the USA.
5 A common finding being that the elasticity of substitution estimated from the first-order condition with respect to labour seems to be systematically higher than that derived from the first-order condition with respect to capital.
6 The benefit of which is that it treats the first-order conditions of a profit-maximizing firm as a system, containing cross-equation parameter constraints, which may fundamentally alleviate the identification of structural parameters as, for example, the elasticity of substitution and technical progress parameters. Thus, the estimation of the whole supply-side system not only contains factor-income share equations, but also an explicit (CES) production function.
7 Note that we scaled the Box–Cox specification by t0 to interpret 
N and K directly as, respectively, the rates of labour- and capital-augmenting technical change at the fixed point.
8 Assuming a specific, albeit flexible, function form for technical progress has the added advantage of circumventing problems related to the non-identification theorem of Diamond et al. (1978).
9 Due to the non-linearity of the CES functional form, sample averages (arithmetic or geometric) need not exactly coincide with the implied fixed point of the underlying empirical CES function. That would be the case only if the functional form is log-linear, i.e. Cobb–Douglas with constant technical growth. Therefore, we capture and measure the possible emergence of such a problem by introducing an additional freely estimated parameter 
(parameter deviates from unity when the estimated CES function deviates from the log-linear Cobb–Douglas case with constant technical growth).
10 See the discussion in McAdam and Willman (2004b).
11 Note that, though we use euro-area data, the data concerns and modelling strategy apply to many large constituent countries (e.g. Germany, France). Our analysis could, therefore, be mechanically performed at the level of individual euro-area country level; we leave this for possible future research.
12 The term 0.03 * DLIB represents the upward correction in the real marginal financing costs in the period of financial regulation as estimated in McAdam and Willman (2004a) for the euro area. DLIB is a hyperbolic level-shift dummy that takes a value of unity in the early 1970s and starts deviating from unity in around 1976 and converges to zero around 1983.
13 From around over 50 per cent in 1970 to around two-thirds at present (the USA has barely changed its sectoral share in services since the 1970s, which was already at around two-thirds of total output).
14 For a detailed discussion on how aggregation across heterogeneous sectors introduces both these components into the aggregated-level relation see McAdam and Willman (2004b).
15 Estimations of non-linear systems can be sensitive to starting parameters. Therefore, to ensure the global outcome, we performed a prior fine-grid search of all parameters (individually and jointly) around broad and plausible ranges. Details are available.
16 This mark-up value is perfectly well in line with the sample average in the underlying data as discussed above (section V(i) United States).
17 This relatively low mark-up may reflect, among other things, the fact that the euro-area data contain imputed housing income and housing stock. Owing to the rent regulations in many European countries, especially in the 1970s and 1980s, the imputed rent component tends to contain a very low or even negative pure profit component.
18 Of course, cross-country comparisons are always hindered by differences in national accounting definitions and practices. For example, in the USA, software is counted as investment, while in the euro area it is more a form of intermediate costs; these differences, in turn, have a bearing on output calculation and investment rating and sentiment. Complete comparability is problematic owing to these underlying data problems.
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