Conversations with Lawrence J. Lau and Peter Temin have helped clarify my thinking on a number of the problems taken up in this paper. Takeshi Amemiya and Donald N. McCloskey suggested improvements upon my statement of several econometric points in an earlier draft. They each have my gratitude, but must be absolved of responsibility for any errors I have committed herein.

¹ J. G. Williamson, “Embodiment, Disembodiment, Learning by Doing, and Returns to Scale in Nineteenth-Century Cotton Textiles,” this Journal. The paper is referred to hereinafter as Williamson, “Embodiment” (1972). My original paper, David, P. A., “Learning by Doing and Tariff Protection: A Reconsideration of the Case of the Ante Bellum United States Cotton Textile Industry,” this Journal, XXX (Sept. 1970), pp. 521–601Google Scholar, is cited hereinafter as David, “Learning” (1970).

² For further discussion, see below, p. 721. This point is conceded by Williamson, “Embodiment” (1972), footnote (15).

³ On some of the more general econometric questions this may involve, cf. Fisher, F. M., A Priori Information and Time Series Analysis: Essays in Economic Theory and Measurement (Amsterdam: North-Holland Publishing Co., 1962)Google Scholar, ch. i. For further discussion of the particular issues confronting the economic historian, cf. David, P. A., “The Future of Econometric History,” in Frontiers of Quantitative Economics, Intriligator, M. D. (ed.), (Amsterdam: North-Holland Publishing Co., 1971), pp. 459–67.Google Scholar In the latter paper, as in this, I find myself in agreement with many of the points that are cogently made by Basmann, R. L., ‘The Role of the Economic Historian in Predictive Testing of Proffered ‘Economic Laws',” Explorations in Entrepreneurial History, 2nd Series, Vol. II, No. 3, 1965.Google Scholar

⁴ David, Cf., “Learning,” (1970), pp. 573–85.Google Scholar An attempt at restating the motivation and methodology of the rather lengthy section of my article entitled “A Priori Considerations and the Choice Between Learning Functions” seems very much in order here, as the fact that both are thoroughly lost from sight in Williamson's discussion is undoubtedly attributable to a lack of sufficient clarity in my original exposition.

⁵ Further, I am now able to cite another historical case, this one for a single integrated cotton textile mill operating in the same period, where the specification of elapsed time as the argument of learning function leads to thoroughly satisfactory regression estimates, which are consistent with those obtained for the multi-mill sample. Cf. David, P. A., “The ‘Horndal Effect’ in Lowell, 1834–56: A Short-Run Learning Curve for Integrated Cotton Textile Mills,” Stanford Research Center in Economic Growth, Memorandum 112, July 1971.Google Scholar

⁶ Cf. David, , “Learning,” (1970)Google Scholar, Table 5. The maximum likelihood estimate cited for â_S + â_L in the text is that obtained from Regression V^c. In the closing section of the original article I may have unwittingly contributed to obfuscating the finding of mild decreasing returns, by presenting the conventional “growth accounting” calculations of the sources of labor productivity increases on the assumption of a constant cost aggregate production function. This was a heuristic strategy decision, intended to spare readers the burden of coping (in the last stages of an already long paper) with the novelty that would be introduced by explicitly exhibiting the effects of mild decreasing returns. Instead, I simply assumed that , chose a maximum estimate of α^° = .5, and computed the parameters used in Table 6 as:

The alternative procedure—which I suppressed—was to employ the smaller estimate of [cf. Ibid., pp. 583–4, n.93, the figure is the mean of the two estimates obtained írom McGouldrick's work], and the absolute magnitudes of the reduced-form parameters as follows:

These latter parameter estimates, as may be seen, turn out to be virtually identical to the values actually employed in constructing Table 6; the only notable difference would be caused by the use of the lower estimate of in place of the 0.5 figure. But it was shown that the important relative magnitudes disclosed by the calculations in Table 6 must be invariant with respect to the absolute value selected for and such differences as the selection of would occasion in the absolute levels of the entries in the table were explicitly reported [Ibid., pp. 597–8, n107] in the accompanying discussion. In short, explicitly acknowledging the presence of decreasing returns does not affect the conclusions I reached with regard to the sources of labor productivity growth, although it does matter in other respects as we shall see.

⁷ As an explanation this is quite consistent with the other propositions I advanced —concerning the position of those textile manufacturing firms who, like the Blackstone Co., found themselves the dominant employers in comparatively thin local labor markets. Although they appear not to have enjoyed any significant economies of scale, by the 1830's their operations nevertheless had been expanded to a point at which rising marginal labor costs were encountered. Such enterprises would thus have continued to derive some local market monopsony rents, even if the inelasticity of the New England regional labor supply meant that the entrance of new textile producers tended to force up the real cost of labor throughout the industry.

⁸ “Embodiment” (1972), p. 697. Elsewhere it is said that “Although Professor David feels able to discriminate between these two learning models, we have been unable to do so given our alternative model specification.” This statement is literally true, but even more disingenuous than the one cited above; “given” Williamson's specification of constant returns to scale and perfectly competitive factor markets, it is obvious that one could hardly proceed to construct the argument which permitted me to arrive at a choice between the regression equations based upon the alternative learning hypotheses.

⁹ The likelihood ratio tests would not correspond to the more familiar F-ratio tests when the unconstrained alternative was the version of my regression model estimated (cf. David, , “Learning” (1970), p. 569Google Scholar, Regression V^c) from variables subjected to the transformation designed to purge the residuals of autocorrelation. In view of Williamson's willingness to impose homogeneity specifications on intuitive grounds, and the possibility of adopting a Bayesian approach on such matters—which is taken up in the next section—it is appropriate to note that any Bayesian strategy that Williamson might have adopted would call for a likelihood ratio test. On these tests, and the statistical theorem just noted, cf. Mood, A. M. and Graybill, F. A., Introduction to the Theory of Statistics (New York: McGraw-Hill Book Co., 1963)Google Scholar, Paras. 12.1–12.9.

¹⁰ Bolles, Albert S., Industrial History of the United States (3rd ed.; Norwich, Conn.: The Henry Bill Publishing Co., 1881), p. 418.Google Scholar As Bolles explicitly states (p. 417) that the source of the figures given for “1870” is “the census-report of that year,” one may reasonably ask why Bolles is to be accepted as a more reliable authority on the findings of the Census than the Census Office itself. It would not matter were the two in agreement. But according to the summary tabulation of final reports for the U.S. cotton textile industry, the share of wages in value added at the date of the Ninth Census (1870) was 0.508, leaving the gross return to capital 49.2 percent of value added rather than the 40.7 percent cited by Williamson. Cf. U.S. Census Office, Twelfth Census of the U.S., 1900, Manufactures: Textiles (Washington, D.C.; GPO, 1902), pp. 54–5.Google Scholar

¹¹ This statement is true not only of the coefficients of In(S/L) presented in “Embodiment” (1972), Table 2, but also of the coefficients in In[Z(θ)/L], the “effective capital’-labor ratio, presented in Tables 3 and 4. The industry-wide shares S_K for 1850 and 1860 were, respectively, .434 and .627, so that all â_S estimates in Table 2 for 1834–60 and 1834–49 would also lie below the minimum share figure for 1850. Cf. U.S. Twelfth Census, 1900, Manufactures: Textiles, pp. 54–5.

¹² Cf. Parker, William N., “From Old to New to Old in Economic History,” this Journal, XXXI, no. 1 (March 1971), 6.Google Scholar

¹³ Jorgenson, D. W., “The Embodiment Hypothesis,” Journal of Political Economy, LXXIV, No. 1 (Feb. 1966), 1–17.CrossRefGoogle Scholar

¹⁴ Cf. Gibb, G. S., The Saco-Lowell Shops, Textile Machinery Building in New England, 1813–1849 (Cambridge, Mass.: Harvard University Press, 1950)CrossRefGoogle Scholar; Navin, T. R., The Whitin Machine Works Since 1831 (Cambridge, Mass.: Harvard University Press, 1950), esp. chs. i, ii.CrossRefGoogle Scholar

¹⁵ It is easy to see that Williamson's recourse to subdivision of the data into observations for two (strangely intersecting) periods, 1834–1849 and 1845–1860, provides no real escape from this dilemma. To avoid the objection raised against specifying a constant θ within each of these 15-year intervals, further sub-periodization would be required, until the degrees of freedom required for the regression estimation became exhausted—impaling the investigation on the other (Jorgensonian) horn of the dilemma.

¹⁶ Abramovitz, M. [”Economic Growth in the United States,” American Economic Review, LII, No. 4 (Sept. 1962), 762–82.]Google Scholar drew attention to the need for such inquiries. Unhappily Jorgenson's response [”Embodiment,” (1966) ] had the unintended but nonetheless perverse consequence of discouraging contemporary and historical research on the question.

¹⁷ When the realities of maintenance and component repair of machinery are recognized, the sharp diehotomization of technical progress into literally “embodied” and “disembodied” innovations is quickly seen to break down. The only sensible way to preserve the meaningfulness of the “embodiment hypotheses” is, therefore, to associate it exclusively with the proposition that (apart from physical wear and tear not rectified by normal maintenance) the operating characteristics of machines of a given vintage remain fixed throughout their service life.

¹⁸ McGouldrick, P. F., New England Textiles in the Nineteenth Century: Profits and Investment (Cambridge, Mass.: Harvard University Press, 1968)Google Scholar, Appendix B, passim; especially Table 22.

¹⁹ Cf. David, P. A., “The ‘Horndal Effect’ in Lowell, 1834–1856: A Short-run Learning Curve for Integrated Cotton Textile Mills,” Center for Research in Economic Growth, Stanford University, Memorandum No. 112, July 1971.Google Scholar

²⁰ For an introduction to the applications of Bayesian statistics, cf. Schlaifer, R., Introduction to Statistics for Business Decisions (New York: McGraw-Hill Book Co., 1961).Google Scholar

²¹ Detailed discussion of the nature of the measurement errors, and the attendant econometric woes, is relegated to the appendix, since these troubles which beset Williamson's work may be said to spring from sources other than a mishandling of prior information—the malady with which this essay is primarily concerned.

²² Cf. David, , “Learning,” (1970), p. 593Google Scholar, esp. n. 102, for the proof of the exclusive dependence of these relative magnitudes upon the regression estimates of the reduced-form parameters α_s and λ obtained directly from Regression V^e. See above, n.6, on the estimates of absolute importance provided by the original article's accounting of the sources of productivity growth.

²³ The only comparable wage rate figures for male and female cotton textile workers that Lebergott [”Wage Trends, (1960), p. 465] is able to present for the years 1850 and 1860 are those extracted from the original census reports of the great Merrimac Mill. The returns for that gargantuan establishment, which already employed upwards of 1,400 workers in 1833, show the relative average wage of the female operatives declining from .576 to .562 over the course of the 1850's. It should be noted that the Merrimac relative wage (ω_f) for 1850 approximates, but does not agree exactly with the national relative wage estimate appearing in Table 1; thus, using the national employment share figures in conjunction with the Merrimac relative wage gives rise to a slightly deviant 1850 female wage share figure s_f, = .506, which falls to s_f = .475 by 1860. Accepting the set of average rates of change computed from these all too fragile estimates, the formula given by equation (5) tells us that if the 1850's witnessed any change in the quality index z worth mentioning, it was a deterioration and not an improvement:

²⁴ Actually Williamson's procedure is not accurately described by his equation (2) [”Embodiment,” p. 695]; he does not apply the quality improvement factor ( 1 + θ)^t to all annual changes in the variable S, but rather only to the positive increments in the latter series. This can be confirmed by comparison of the first and any subsequent column of his Table 1.

²⁵ Cf. Williamson, , “Embodiment” (1972), p. 691–692Google Scholar; also, Williamson, J. G., “Optimal Replacement of Capital Goods: The Early New England and British Textile Firm,” Journal of Political Economy, LXXIX (Nov./Dec, 1971), 1320–34.CrossRefGoogle Scholar

²⁶ One possible mitigating condition is hinted at in Williamson's [fn. 10] reference to McGouldrick's finding that in the short-run replacement investments and capacity expansion tended to occur together. But as the McGouldrick data presented by Williamson [fn. 10] shows, the relationship of the two hardly appears to have been sufficiently stable to permit the substitution of net investment as a proxy for gross investment.

²⁷ For example, consider abrupt rise in S = Z ( θ = 0) between 1840–41: according to Williamson's Table A.1, col. 3, the effective stock (Z _.01) rose by 11.1%, from 123.5 to 137.9, whereas, from the first column of the table we can see that the major part of the rise in S in 1841 represented a return towards the level of utilization that had been attained in 1837. The change measured by Williamson is (136.5 — 123.1) (1.01)⁷ = 14.4, but of the difference (136.5 - 123.1) only 4.1 ( = 136.5 - 132.4) represented an increase beyond the previous peak value of S, in 1837. Thus, on the hypothesis that quality improvements affected only the investment goods that expanded capacity in the industry, Williamson ought to have computed the increase of the effective capital input (Z_.01) as [(9.3) + (4.1) (1 + .01 )⁷] = 13.7, rather than as 14.4.

²⁸ The familiar result on unsystematic errors in (exogenous) variables in a univariate regression is unfortunately uninstructive in informing us of the direction of the bias of the regression coefficient on In[Z(θ)/L] in Williamson's formulation, although there is some presumption of a bias downwards. Cf., e.g., Malinvaud, E., Statistical Methods of Econometrics, (Amsterdam: North-Holland Publishing Co., 1966), pp. 331–5.Google Scholar The coefficients on the experience indexes (either cumulated output or elapsed time) will also be inconsistent and biased estimates, and may well be similarly subject to a downward bias—since there is some, albeit weak, positive covariation between the natural logarithm of (S/L) and both In(Q) and In (T) —but we cannot be certain.