Before anything else, I wish to acknowledge my indebtedness to Robert B. Zevin, whose vet unpublished work applying a long-run learning function to data on an ante-bellum cotton textile firm inspired the econometric research reported in this article. Professor Zevin may be held responsible for having started this hare, though not for the course it has run. The subsequent direction taken by my thoughts on the subject of learning and tariff policy was inexorably (but pleasurably) influenced by many conversations with my colleagues Emile Despres and Ronald McKinnon. They ought not, however, be blamed for any of my failures in faithfully absorbing their wisdom on the subject. The empirical investigation of learning effects was carried out in conjunction with the Stanford-SSRC Study of Economic Growth in the U.S., and benefited immeasurably from the insight of my colleague in that project, Moses Abramovitz. Finally, I wish to acknowledge the able computational assistance Harry Cleaver provided in connection with this study.

¹ Cf. Taussig, F. W., The Tariff History of the United States (8th ed.; New York: G. P. Putnam's Sons, 1931)Google Scholar, especially p. 2.

² Ibid., pp. 2–3.

³ Cf., Ibid., pp. 25–36, 135–42 for Taussig's discussion of the industry in the antebellum period. (The quote appears on p. 25.)

⁴ Ibid., p. 136.

⁵ On the eve of the Civil War the (f.o.b.) value of domestic exports accounted for something on the order of 8–10 percent of the gross value of U.S. cotton goods production. Cf. U.S. Census Bureau, Eighth Census of the United States, (1860), “Manufactures,” pp. 733–42, for gross value of product, and Taussig, Tariff History, p. 142 for 1859–60 export values from the Reports of the Secretary of the Treasury on Commerce and Baines, Navigation. E. (A History of the Cotton Manufacture in Great Britain [London, 1835], pp. 509–10Google Scholar) concluded, from a comparison of the costs of manufacturing cotton goods in the U.S. and England in 1832: “It may be said that Americans are capable of rivaling the English in coarse and stout manufactures, … especially in an article called ‘domestics,’ which they consume largely and export to some extent; but that in all other kinds of goods, all of which require either fine spinning or hand-loom weaving, the English possess and must long continue to possess a great superiority; in other words, ‘the Americans cannot economically produce fine manufactures.’”

⁶ Cf. Lerner, A. P., “The Symmetry Between Import and Export Taxes,” Economica III (Aug. 1936), pp. 308–13;Google Scholar also, McKinnon, R. I., “Intermediate Products and Differential Tariffs: A Generalization of Lerner's Symmetry Theorem,” Quarterly Journal of Economics LXXX (Nov. 1966), pp. 584–615CrossRefGoogle Scholar.

⁷ Taussig, Tariff History, p. 142.

⁸ Cf., e.g., North, D. C., Growth and Welfare in the American Past (Englewood Cliffs, N.J.: Prentice-Hall, 1966), p. 78;Google ScholarFite, G. C. and Reese, J. E., An Economic History of the United States (2nd ed.; Boston: Houghton-Mifflin, 1965), pp. 244–45.Google Scholar However, Taussig's views on the sources of the growth of the U.S. pig iron industry prior to 1860, including the role he assigned the tariff changes of the 1840's, have recently received some consideration. Cf. Temin, Peter, Iron and Steel in Nineteenth Century America (Cambridge: M.I.T. Press, 1964);Google ScholarFogel, Robert W. and Engerman, Stanley L., “A Model for the Explanation of Industrial Expansion during the Nineteenth Century: With an Application to the American Iron Industry,” Journal of Political Economy, LXXVII, No. 3 (May/June, 1969), 306–28CrossRefGoogle Scholar.

⁹ Cf., e.g., the treatment of the subject by Clark, V. S., History of Manufactures in the United States, I (New York: McGraw-Hill, 1929), 450–53;Google ScholarNorth, D. C., The Economic Growth of the United States, 1790–1860 (Englewood Cliffs, N.J.: Prentice-Hall, 1961), Ch. ixGoogle Scholar, passim.

¹⁰ Cf. Davis, Lance E. and Stettler, H. Louis, “The New England Textile Industry, 1825–60: Trends and Fluctuations,” Output, Employment and Productivity in the United States After 1800, National Bureau of Economic Research Studies in Income and Wealth, XXX, (New York: 1966), esp. 227–32Google Scholar. Davis and Stettler also attribute to economies of scale the differences they observe between average labor productivity in the large integrated firms in the cotton textile industry and average labor productivity in the U.S. industry as a whole during the period c.1830-c.1860. Cf. Ibid., p. 231. This point is discussed below.

¹¹ Zevin, Robert B., “The Use of a ‘Long Run’ Learning Function: With Application to a Massachusetts Cotton Textile Firm, 1823–1860,” Mimeographed for the University of Chicago Workshop in Economic History, November 22, 1968Google Scholar. See below for fuller discussion of Zevin's work.

¹² Cf. Bhagwati, Jagdish, “The Pure Theory of International Trade: A Survey,” Surveys in Economic Theory, II, American Economic Association-Royal Economic Society, (New York: St. Martin's Press, 1967), esp. 214 ffGoogle Scholar.

¹³ The extent of this shift is made fully evident by the recent appearance of a cogently argued piece by Baldwin, Robert E., “The Case Against Infant-Industry Tariff Protection,” Journal of Political Economy, LXXVII, No. 3 (May/June 1969), 295–305CrossRefGoogle Scholar.

¹⁴ This assumes that foresighted entrepreneurs have access to perfect capital markets, and can therefore finance the ‘learning period” during which their costs might remain above those of well-established foreign competitors. Capital market imperfections would give rise, understandably, to a range of inefficiencies in resource allocation; some of those would involve learning and training effects, i.e., the inadequate allocation of resources to activities in which such effects were present, and might be partially rectified by tariff intervention. One aspect of this broad set of problems is given further notice below, but otherwise the discussion of the implications of learning by doing for tariff policy abstracts from the second best situations created by factor market distortions.

¹⁵ On the crucial importance of the existence of externalities to the “modern” infant industry argument, cf., Meade, J. E., Trade and Welfare (London: Oxford University Press, 1955), p. 256;Google ScholarJohnson, H. G., “Optimal Trade Intervention in the Presence of Domestic Distortions,” in R., Baldwin, et al., Trade, Growth and the Balance of Payments, (Amsterdam: North Holland Publishing Co., 1965)Google Scholar; Bhagwati, “Pure Theory of International Trade,” p. 218.

¹⁶ Taussig, Tariff History, pp. 3–4.

¹⁷ In the case of training of workers in a transferable skill, the rational firm will not anticipate being able to go on paying their employees less than their marginal value products in order to recapture a return on the training costs. Instead it will fall to the workers to finance their acquisition of expertise in the new line of production. Quite apart from the problems that would arise if workers had to rely upon imperfect capital markets to finance this investment, it seems appropriate to ask whether it would actually be rational for the workers to invest in such readily transferable— and hence generally obtainable—training. While the first cadre of workers to do so might enjoy quasi-rents on their skill until the growth of the industry, and its trained labor force, multiplied the opportunities for training, is it not to be expected that the falling cost of training will create a situation rapidly approaching that of free entry into the occupation, and that the market value of the original investment would be drastically reduced in this process? If this is the case, the workers who are asked to finance the generally transferable portion of their training are in much the same position as that of pioneer firms who worry about subsequent entrants depriving them of a return upon their initial investment in learning. The question remains, however, as to whether imposing a tariff—rather than directly subsidizing training in transferable skills—is a desirable way of coping with this problem.

¹⁸ Cf. Asher, Harold, Cost Quantity Relationships in the Airframe Industry (Santa Monica: The RAND Corporation, R-291, July 1956);Google Scholar also, Alchian, A., “Reliability of Progress Curves in Airframe Production,’ Econometrica XXXI (Oct. 1963), pp. 679–93CrossRefGoogle Scholar.

¹⁹ Hirsch, Werner Z., “Manufacturing Progress Functions,” Review of Economics and Statistics, XXXIV (May 1952), 143–55CrossRefGoogle Scholar. Cf., also, by the same author, “Firm Progress Ratios,” Econometrica, XXIV (April, 1956), 136–43.

²⁰ Rapping, Leonard, “Learning and World War II Production Functions,” Review of Economics and Statistics, XLVII (Feb., 1965), 81–86CrossRefGoogle Scholar.

²¹ Cf. Arrow, Kenneth J., “The Economic Implications of Learning by Doing,” Review of Economic Studies, XXIX (June, 1962), 155–73CrossRefGoogle Scholar.

²² Cf. Sheshinski, Eytan, “Tests of the Learning by Doing Hypothesis,” Review of Economics and Statistics, XLIX (Nov. 1967), 568–78CrossRefGoogle Scholar.

²³ Cf. Ibid., p. 568 n.l, on the implications for international trade of the existence of “irreversible’ economies of scale, i.e., those which “appear with every act of investment and do not disappear subsequently.”

The econometric findings reported by Sheshinski are based on cross-section observations for manufacturing industries in U.S. states in 1957, and mixed cross-section time series data for selected manufacturing industries in several countries during 1950–1960. The former part of the study takes gross book values of the 1957 capital stock of the industry (in each state) as a measure of cumulated gross investment, without any justification for equating the two quite distinct concepts. Unfortunately, nowhere in the article or the appendix on “Sources of the Data” does Sheshinski describe the derivation of the cumulated output and cumulated gross investment variables used in the cross-section time series study; there is some doubt whether the cumulations begin prior to 1950.

²⁴ Of course, the story of the emergence of specialized machinery producers, like the Saco-Lowell shops and the Whitin Machine Works, from early cotton textile mill machine shops is usually told in terms of the spin-off of activities subject to economies of scale. But perhaps the role of the textile tariff, in inducing new entrants to the industry (and existing smaller mills) to specialize in cloth production and depend upon a few “outside” shops for their equipment, warrants more attention than it has received. Cf. Gibb, G. S., The Saco-Lowell Shops, Textile Machinery Building in New England, 1813–1849 (Cambridge, Mass.: Harvard University Press, 1950);CrossRefGoogle ScholarNavin, T. R., The Whitin Machine Works Since 1831 (Cambridge, Mass.: Harvard University Press, 1950)CrossRefGoogle Scholar, esp. ch. i, ii; D. C. North, The Economic Growth of the United States, pp. 10, 161–62.

²⁵ The importance of finding means of easing the constraints imposed by capital market imperfections in situations where indivisibilities are present in production processes that offer greatly reduced marginal costs has received recent emphasis in discussions of economic development policies by McKinnon, R. I. and Shaw, E. S.. Cf., e.g., R. I. McKinnon, “On Misunderstanding the Capital Constraint in LDC's: The Consequences for Trade Policy” Typewritten Draft, Summer 1969)Google Scholar.

²⁶ This latter stricture applies equally to the proposal that tariff support be provided as a means of permitting firms in the industry to pay higher wages and thereby make it possible for workers to internally finance their training in industrial skills that are transferable, and consequently not financed by rational firms. It is clear that the purpose would be served equally well by other fiscal measures that effected the same net redistribution of income in favor of workers engaged in textile production; welfare theorists would naturally prefer to do this by lump sum grants to the workers involved. Moreover, it is debatable whether raising the domestic price of the industry's product by means of tariffs would have the desired result of helping to overcome the problem of financing labor-force training when capital markets are imperfect. Quite possibly the higher product prices would simply induce firms to expand their employment of untrained (lower marginal productivity) laborers. Cf. Baldwin, “Tariff Protection,” p. 301, on the last style of objection to tariffs.

²⁷ The following discussion is largely based upon the research of Lance Davis, E., “Stock Ownership in the Early New England Textile Industry,” The Business History Review, XXXII, No. 2 (Summer 1958), 204–22;CrossRefGoogle Scholar “The New England Textile Mills and the Capital Markets: A Study of Industrial Borrowing 1840–1860,” Journal of Economic History, XX (Mar. 1960), 1–30. Cf. also, Federal Reserve Bank of Boston, A History of Investment Banking in New England, Annual Report for 1960, (Boston, 1961), ch. i. For further material regarding the balance sheet positions and investment financing practices of the cotton textile companies whose production operations are the focus of attention in the present study, cf. McGouldrick, Paul F., New England Textiles in the Nineteenth Century, Profits and Investment (Cambridge, Mass., 1968), pp. 14–18, 134–38Google Scholar.

²⁸ This accepts L. E. Davis’ conclusion that respect for the Massachusetts usury law's 6 percent ceiling, on the part of the quasi-public, philanthropic intermediaries, had the effect of segmenting the market for loanable funds and Keeping the longterm rate persistently below the short-term interest rate during the 1840's and 1850's. Vatter, Barbara (“Industrial Borrowing by the New England Textile Mills, 1840–1860,” Journal of Economic History, XXI [June 1961], 216–21)CrossRefGoogle Scholar, on the other hand, has argued that interlocking directorates between financial and textile firms were primarily responsible for the low rates charged the latter by the former. But, as Davis, (“Mrs. Vatter on Industrial Borrowing: A Reply,” Journal of Economic History, XXI [June 1961], 222–26)CrossRefGoogle Scholar has observed, there is no clear evidence of systematic customer discrimination favoring the textile industry in the rates charged by either commercial banks, savings institutions, or insurance companies such as Massachusetts Hospital. It should be noted, nevertheless, that with the long-term rate legally held below the free market equilibrium there would be ample opportunity, in the rationing of funds by the non-bank intermediaries, for the web of interlocking directorates to protect the interests of the textile mill owners at the expense of other supplicants for loans.

²⁹ Cf. Navin, Whitin Machine Works, pp. 10, 206, 269. Wilkinson was Slater's American-born brother-in-law, and an exceptionally clever blacksmith.

³⁰ It is likely that in the (northern) section of the industry where integrated spinning and weaving mills were more predominant, the Boston Manufacturing Company's original mills at Waltham fulfilled a parallel training function. Moreover, it seems plausible that while the illustration supplied in the text concerns the training of machinery-builders, the pioneer firms also played a significant role in the education of cadres of future superintendents and mill managers. Close historical studies of the latter type of training effects would be of considerable interest, but still remain to be done in the case of the early cotton textile industry.

³¹ Rapping (“Learning”), Sheshinski (“Tests of Learning By Doing”), and Zevin (unpublished work); all have formulated tests of the hypothesis in long run terms. The general statement of the production model employed here, with some modifications, follows that given by Sheshinski, “Tests of Learning by Doing” pp. 569–70.

³² Of course, one might alternatively specify that there were decreasing returns to experience, so that the level of efficiency in any activity rose—perhaps along some logistic or Gompertz function—towards an upper asymptote. With experience rising at a constant rate under such circumstances, the rate of efficiency growth would fall continuously, reaching zero in the limit. As is shown below, it is quite possible to generate the latter efficiency-path with an “unbounded” form of learning function, such as that in equation (4), by a suitable choice of the experience index, G. We shall therefore avoid the added econometric complexities that would accompany experimentation with alternative algebraic forms for the learning function.

³³ If the index of productivity is Π(τ)/Π(0), the “learning curve” can be interpreted as implying that

where C(τ)/C(0) is the index of costs at constant input prices. Thus, doubling cumulated output, we have: (0.5).33 ∼ .80. In some studies with production facilities held fixed the productivity index has related only to labor inputs, whereas, in others where physical facilities were also changed, the productivity meausres were of the total, rather than the partial type. Cf. the estimates from Hirsch, “Firm Progress Ratios.” Rapping, “Learning,” reports significant learning coefficients of 0.29 using cumulated output of Liberty ships (including current production), and coefficients of 0.23 to 0.34 using cumulated Liberty ship output including only half the current year's production. The original observation of a “learning curve” in the case of airframes production—relating only to the behavior of labor costs—served to define a learning curve as one in which the coefficient λ was below unity. Cf. Asher, Cost Quantity Relationships; also, A. Alchian, ‘The Reliability of Progress Curves.”

³⁴ Cf. Zevin (unpublished work), pp. 21–22.

³⁵ After this section had been drafted, my attention was drawn to a very recent paper by Fellner, William, “Specific Interpretations of Learning by Doing,” Journal of Economic Theory, I (August 1969), 119–40CrossRefGoogle Scholar. In this article Fellner (p. 124) distinguishes two “versions” of the learning hypothesis:

“Why assume that experience in the relevant sense is acquired by doing more than one has so far done, regardless of the length of time it takes to do more (Version A); and why not assume that experience is acquired by doing it longer, regardless of the steepness of the rise of cumulated output (Version B)?”

Fellner's Version B is, of course, the cumulated time formulation suggested—with an argument for its plausibility—by R. B. Zevin, to whom I have here awarded (unclaimed) priority.

³⁶ Cf. Davis and Stettler, “The New England Textile Industry,” pp. 218–19; Zevin, (unpublished work), pp. 10–11. Paul F. McGouldrick also has employed running yardage measures (cf. McGouldrick's “Comment” in Output, Employment and Productivity in the U.S. After 1800 (N.B.E.R. Income and Wealth Conference, Vol. XXX), p. 240; Layer, Robert G. (Earnings of Cotton Mill Operatives, 1825–1914, [Cambridge, Mass.: Committee on Research in Economic History, 1955])Google Scholar maintained that pounds of cloth provided a superior output measure. Poundage figures nominally take account of count differences but ignore width variations in cloth.

³⁷ At least this is the case for the sample of mills for which both yardage and poundage figures were obtained by Davis and Stettler, (“The New England Textile Industry,” p. 219, n. 9). The latter source (Ibid., Table A-2, p. 238) provides annual output estimates in standard yards of 14s × 14s, 48 × 48, 36-inch wide brown sheeting, and corresponding heterogenous yardages for a group of nine firms engaged primarily in manufacturing low count goods: Hamilton, Boston, Suffolk, Tremont, Lawrence, Nashua, Naumkeag, Jackson, and Pepperill. The (Y) output series employed here (see Tables 1 and 2) pertains to the first six firms of this group.

³⁸ Cf. underlying figures in U.S. Eighth Census, “Manufactures,” pp. 733–42. Detailed costs accounts, available in textile company manuscripts from the era under review, disclose that cotton costs were over 90 percent of the cost of all purchased materials. Cf., McGouldrick, New England Textiles, p. 144. It should be noted that in the nineteenth century power was treated, by and large, as a capital expense, rather than as a purchased item, in the New England textile company accounts.

³⁹ Davis and Stettler (“The New England Textile Industry,” p. 218, n. 8) refer to “the rather constant proportion between cotton input and cloth output (regardless of the quality of output).” Zevin (unpublished work, p. 16, n. 10) reports that in the case of the Blackstone Manufacturing Co., the ratio or the weight of cloth (Y′) to (C) the weight of raw cotton, increased only very slightly over the period 1823 to 1860, from 0.87 to 0.92, i.e., C(τ) = c(0) Y′(τ) exp {—.0015 τ}.

⁴⁰ Cf. McGouldrick, New England Textiles, p. 145, for use of a fixed yards-perpound conversion factor (4.36 = 1/k) which would be roughly appropriate for the average count level of the cloth produced by the New England mills covered in the present inquiry.

⁴¹ Note that it is actually not necessary that (m) be strictly constant. So long as the variations in m are neither serially correlated nor correlated with any of the other arguments of the production function for Y*, one can replace C by ckY* and then substitute qY for Y*, obtaining a least-squares regression model in which the variation around E(mτ) = m will manifest itself, acceptably, as errors in the dependent variable: (ln Y_τ). Use is made of this below.

⁴² Zevin (unpublished work, works with output per worker measures of labor productivity, and explicitly assumes the ratio of women and children employed to the total number of textile operatives in the Blackstone Manufacturing Co. labor force remained constant at its average 1823–60 level of .88. Davis and Stettler (“The New England Textile Industry,” p. 221) mention changes in hours of work but conduct their discussion of labor productivity on the basis of movements of output per worker.

⁴³ In the short-run the problem is complicated by the fact that whereas initially women and children were predominant in the labor force of the spinning and weaving departments, adult males were employed within the mills on the construction and installation of machinery, as well as in activities more immediately connected with current cloth production—such as carding, dressing, or machinery maintenance. Variations in the extent of the capital formation activity taking place within the mills, which would not of course be registered in the movements of (cloth) output at the time, would thus be likely to cause abrupt alterations in the ratio of adult males to women and children employed. For this reason such short-run sex-composition shifts are probably best ignored in a study focusing upon cloth production (rather than the joint activities of cloth production and machine construction). That, essentially, is the approach adopted by Zevin (unpublished work, pp. 9–10) who constructs an index of labor inputs based on the number of women and children alone, assuming that the relationship between those workers and males engaged in current cloth production remained constant. Yet this solution to the short-run problem unfortunately leaves out of account the effects of longer-run alterations in the make-up of the cotton textile work force, and it is with the latter that we shall here be concerned.

⁴⁴ Cf. Lebergott, Stanley, Manpower in Economic Growth (New York: McGraw-Hill, 1964), p. 48Google Scholar, for the evidence cited in the text. Confirmatory data is available in Layer, Earnings of Cotton Mill Operatives, p. 43.

⁴⁵ The chronology of cyclical fluctuations in the New England textile industry prepared by Davis and Stettler (“The New England Textile Industry,” Table 6, p. 224) locates both 1830 and 1860 as falling between troughs and peaks, whereas 1844 is a peak year. If the length of the workday tended to vary positively with the movement of output over the cycle, the mid-point to peak change (1830–44) would understate the secular rate of fall in hours, while the cyclical peak to mid-point change (1844–60) would tend to overstate it. Note, then, that from the data cited in the text, we may compute the following rates of change in the duration of the workday:

⁴⁶ Cf. Lebergott, Stanley0, “Wage Trends, 1800–1900,” in Trends in the American Economy in the Nineteenth Century, N.B.E.R. Income and Wealth Conference, XXIV (Princeton, N.J.: Princeton University Press, 1960), pp. 459–60Google Scholar. On the quite distinct problem of short-run variations in sex-composition, see above, p. 30, footnote 42.

⁴⁷ Cf. Lebergott, “Wage Trends, 1800–1900,” p. 460.

⁴⁸ The calculation is made as follows: Let μ represent the ratio of male workers in 1900 to male workers in 1832, and let θ be the corresponding ratio for female workers. As females represented 0.613 of the Massachusetts cotton textile mills’ workers in 1832 (according to the figure already cited in the text for the ratio of females to males at that date), the following equation gives the relation between μ and θ required to maintain the size of the total work force, in numbers of workers, unchanged between 1832 and 1900:

But, from the change in the female-male worker ratio, we also know that over the 1832–1900 interval,

Solving these two equations yields: μ* = 1.358 and θ* = 0.781.

Now, let Ƶ be an index of the ‘quality’ of the average worker in the industry. The proportional growth of standard labor inputs per worker employed is thus:

.

Since the elasticity is estimated as the male share in the cotton textile wage-bill, S_m = 0.6, we find

which corresponds to an average rate of growth of worker quality of β_L = .0112 per annum over the period 1832–1900.

⁴⁹ Taussig thought that spindles were “the best single indication of the extent and growth of such an industry as cotton manufacture” (Some Aspects of the Tariff Question [Cambridge, Mass.: Harvard University Press, 1915], p 265)Google Scholar, but the relationship between spindles and output of cloth in the U.S. industry was far from fixed, even over the long run. Cf. Davis and Stettler, “The New England Textile Industry,” pp. 227 ff., for discussion of this point, and also the data assembled in Appendix Table I below.

⁵⁰ Cf., e.g., Zevin (unpublished work), p. 10. Switching between count levels would unbalance the machine stock (and the labor force employed) within a given integrated mill, because the number of machines required in the opening and carding department per spindle, and the number of looms required per spindle, were both derceasing functions of the yarn count. The number of looms per spindle also decreased with the number of twists per inch. (Cf. McGouldrick, New England Textiles, p. 29.) These considerations served to narrowly restrict the range of counts of the cloth produced by a single integrated mill. But by the same token, for cloth of a given count and twist, or for a standard mix of different-grade fabrics produced by a group of integrated mills, the number of spindles serves as an adequate index of the entire complement of machinery employed.

⁵¹ Cotton textile firms recorded the number of spindles operated in each mill on a daily basis, and such information has been used by Zevin (unpublished work, p. 10) to secure an index of annual capital services flows for the Blackstone Manufacturing Co. Davis and Stettler do not explicitly state that their estimates of annual cloth output per spindle for a sample of six firms (which is utilized in the present study) pertain to spindles in operation, but the spindlage figures that can be derived from those partial productivity data and the corresponding aggregate output series (see Appendix Table I below for sources) exhibit a pattern of short-period fluctuations which clearly reveal that they cannot refer to the number of spindles in place. For example, the estimates of S (for the six-firm group described in Appendix Table I, below) show a drop of 7 percent between 1837 and 1839, and of 16 percent between 1853 and 1854—both movements being very abrupt and coinciding with business recessions.

⁵² On the number of normal operating days (implied by the six-day work week), cf. Lebergott, “Wage Trends, 1800–1900,” p. 479. Davis and Stettler (“The New England Textile Industry,” p. 227) comment on the reduction of the workday in this connection, and report that when their output per spindle estimates (Ibid., Table 8) are adjusted to a common workday—an adjustment whose details are not described—the 13 percent decline in output per spindle recorded in the decacle following 1840 is no longer observable.

⁵³ I rather suspect this latter position would appeal to Jorgenson, D. W. and Griliches, Z., who, in their study (“The Explanation of Productivity Change,” Review of Economic Studies, XXXIV [July 1967], pp. 249–83)CrossRefGoogle Scholar of the post-World War II U.S. economy, undertake to adjust capital input measures for secular alterations in equipment utilization as reflected in the changing relationship between electric motor capacity and electricity consumption for electric motors (in manufacturing industries).

⁵⁴ Cf. Gibb, The Saco-Lowell Shops, pp. 76–80; Davis and Stettler (“The New England Textile Industry,” p. 230) point out that changes in average operating speeds within the industry continued to disturb the observed spindle-output ratio in the years following 1828, which form the period under examination here.

⁵⁵ With the Cobb-Douglas form all input-augmenting technical changes can be expressed as equivalent Hicks-neutral efficiency improvements. Thus, in equation (13), the compound coefficient (a*_s β_s) would take the place of the rate of exogenous “disembodied” technical change, β, in equation (3).

⁵⁶ Cf. Davis and Stettler, “The New England Textile Industry,” especially pp. 231, 234–36. The labor input and spindle input estimates, (L) and (S), respectively, have to be derived from the partial productivity series (Y/L) and (Y/S) presented by Davis and Stettler, and an aggregation or company production estimates into the corresponding yardage output (Y) of the sample group. For convenience, the series obtained from this source are presented in the Appendix Table I below, along with the corresponding experience indexes, (Q) and (T).

⁵⁷ Cf. Davis and Stettler, “The New England Textile Industry,” Table A-2, p. 238, for the number of mills in their sample of firms producing low-count cloth. Note that Jackson (whose recorded annual yardage output did not increase notably in the decade after 1833) had accounted for one mill when it entered in 1832 (Ibid., Table A-l, pp. 234 ff.), whereas by 1843 Naumkeag and Pepperill did not yet have mills in production. The remaining six firms, and the implied number of mills in 1843 (i.e., 17), constitute the sample group under discussion here. Brief sketches of the organizational histories, and the characteristics of the operations of the individual companies are available in McGouldrick, New England Textiles, Appendix A, pp. 219–21.

⁵⁸ Cf. Davis and Stettler, “The New England Textile Industry,” pp. 230–31. As pointed out in the following footnote, however, the bias on this score cannot have been large.

⁵⁹ The implication is that were the data from the bottom two panels of Table 2 to be made the basis of a scatter-diagram showing the relationship between labor productivity (on the ordinate) and capital intensity, the observations relating to the sample would have to be displaced (vis-a-vis the rest of the points in the scatter) rather far to the right but only slightly upwards. The small upward displacement would take into account the approximate 15 percent understatement of the relative labor productivity standing of the sample firms, resulting from Davis and Stettler's assumption of a work-year of 265 days (rather than the conventional 311 days) in converting the output per man-day figures for the sample into output per man-year—the latter being the basis on which the census labor productivity data for Massachusetts and the U.S. are available.

⁶⁰ Cf. Zevin (unpublished work), pp. 8–9. See Table 3, below for direct comparisons with Zevin's regression results.

⁶¹ The learning coefficients, unlike the exponents of the labor and capital and raw materials inputs in Cobb-Douglas functions, may be subject to an aggregation bias even under conditions of constant returns. Furthermore, there is no necessity for the firm and group learning coefficients to be identical even when the rest of the production function is found to be the same, and account is taken of any aggregation bias. These matters receive fuller attention in the text below.

⁶² It should be understood that the entries in the third column of Table I relate not to the strict age of the companies (commencing with the year of their respective organizations), but instead give the number of years of recorded production experience. Inspection of the output levels in the first years for which figures could be obtained by Davis and Stettler—as shown in Table 1—makes it evident (upon comparison with initial production figures of other firms and subsequent output levels for the firms in question) that while there may be errors in the base year (1834) estimates of Q and T, due to missing initial production records and fractional years of experience, such errors cannot be very serious. For further details regarding approximate dates of incorporation and the inititation of production, cf. McGouldricK, New England Textiles, p. 4 (Table 1) and Appendix A.

⁶³ As can be seen from the t-statistics (given in parentheses beneath the respective regression coefficients) in Table 3, the coefficient a₂ in R-I: Blackstone is significantly different from zero at the 1 percent error level. But what is relevant is whether the labor input coefficient in the production function, a_L = (1 — a₂), is statistically significant. Thus, we must ask whether or not it is possible to reject the null hypotheses H_o:a₂ = 1. The t-statistics computed for this test in the case of the Blackstone regression equations are:

Single and double asterisks indicate rejection at the 5 percent and 1 percent levels, respectively. From Table 3 it is readily seen that in all the regression equations estimated from the sample group data, the same null hypotheses can be rejected easily at the 1 percent level. The appropriate t-statistics for the test of (H_o:a₂ = 1) may be readily computed from information in Table 3, since

where ta is the statistic appearing in parentheses below the regression coefficient of (lnL) in Table 3. All significance levels cited here refer to two-tail tests.

⁶⁴ Specifically, the coefficients of (lnQ) in equations R-II and R-IV of Table 3, and the coefficients of (lIn T) in equations R-III and R-V are all found to be significantly different from zero at the 99 percent confidence level. This is true not only of the equations fitted to the six-firm sample data, but holds equally for Zevin's estimates based on the Blackstone data, at least for those cases (R-II, R-IV and R-V) in which corresponding regressions are reported. See Table 3. As the preceding footnote points out, the results of t-tests of the null hypothesis that the coefficient λ is less than unity may be read indirectly from the t-statistics reported in the table, but it should be remembered that in this case the appropriate level of significance will be that indicated for a single-tail test. On this basis the null hypothesis H_o:λ1 can be easily rejected for every coefficient of (ln Q or (ln T) that appears in Table 3.

⁶⁵ It is unfortunately not possible to make meaningful statements about the statistical significance of the implied values of and presented by Table 4, since the distributions of the associated errors of estimate are quite unknown.

⁶⁶ Zevin's paper (unpublished work), Table I, does not report results of any tests for the presence of significant autocorrelation in the residuals of the equations he estimated from the Blackstone data. The discussion of this matter here is consequently confined to the regressions for the six-firm sample in Table 3.

⁶⁷ Cf. Johnston, J., Econometric Methods (New York: McGraw-Hill, 1963), pp. 194–95;Google ScholarMalinvaud, E., Statistical Methods of Econometrics (Chicago: Rand McNally, 1966), pp. 420 ffGoogle Scholar.

⁶⁸ With 22 degrees of freedom, and three continuous independent variables, the 5 percent significance points of D-W are 1.05 — 1.66. Note that the D-W corresponding to R-V in Table 3 falls as far short of the upper 5 percent point (1.66) as does the D-W for any of the four equations with significant learning effects.

⁶⁹ Note that the test appropriate to the statement made in the text is a single-tail t-test. Although the null hypothesis a₁, = 0 cannot be rejected on a two-tail test at the 95 percent confidence level in the case of R-III, Table 3 indicates that this more exacting test is passed by the a₁-coefficients of the other variants of the learning model fitted to the sample group data.

⁷⁰ See footnote 1, above, for the results of tests of this hypothesis.

⁷¹ To prove that H_o(l) = H_o(2): [a₁ − a₇ = 0], we may begin by noting from (15a) that

However, from (14b) we also know that

If we accept the premise that is positive but less than unity, rejection of H_o(2) therefore must imply rejection of H_o(l).

⁷² From Table 3 it may be seen that the sum of the reduced-form parameter estimates and , yielded by the Blackstone regressions is 0.9185 for R-II, 0.9045 for R-IV, and 0.9192 for R-V. Zevin (unpublished work, p. 12) reports only the result of a test of the constant returns hypothesis for the model estimated by R-IV, but it is clear that the other two regressions tell the same story.

⁷³ As pointed out earlier, this is a terribly stringent requirement which presupposes that Blackstone was indistinguishable from the representative member of the six-firm sample. Yet, it should be noted that Zevin's estimates for the Blackstone production function refer to the period 1823–60, which is not quite the same interval as that for which the sample group estimates are available. Consequently one ought not make too much of the fact that while the ) coefficients for Blackstone and the sample group are closest in R-V—0.3749 v. 0.2259—they are still not statistically identical like the corresponding estimates of (=1 —a₂) in that equation. The significance test referred to here is a t-test of the significance of the difference between the regression coefficients from the two R-V equations, using the pooled estimate of the standard error computed from:

where N_Blackstone = 33 and N_Group = 22 represent the respective number of degrees of freedom associated with the estimated errors of the regression coefficients.

⁷⁴ As far as Durbin-Wateon statistics go, the differences between R-II and R-III or between R-IV and R-V are negligible, and, in any case the slightly higher degree of autocorrelation in the residuals of R-III and R-V is not worth worrying about—as has been shown by the comparison of R-Vc with R-V in Table 3.

⁷⁵ Zevin (unpublished work), Table I. For the equation cited here slightly lower than that found when the ln Q-term is deleted. Cf. Table 3, R-V: Blackstone.

⁷⁶ The simple correlation coefficient between ln(Q) and ln(T) for the sample group (1834–60) is R = .999.

⁷⁷ The magnitude of Ξ, and the difference between the constrained and unconstrained estimates will not be of concern here. We have already noted that with respect to the inferred values of the underlying structural parameters β_s and β_L, Table 4 (above) reveals there is little to choose between R-IV and R-V. Moreover, none of the relationships between and suggested by the regression equations can be dismissed as implausible. It is therefore necessary to look elsewhere for guidance.

⁷⁸ Cf. Zevin (unpublished work), Table II, p. 13.

⁷⁹ Under conditions of perfect competition in all (product and factor) markets, labor and capital would receive an equilibrium rate of remuneration equal to their respective marginal value productivities, and the elasticities of output with respect to these inputs would exactly equal the shares of the total product each received. But the equality of relative shares (s_i) with the relative “elasticity coefficients, (α_L) /(α_L + α_s) = (S_L)/(S_L + S_s), requires only that the real rates, of factor remuneration be in uniform proportion to their respective marginal physical productivities. The latter condition is compatible with equal degrees of imperfection in all factor markets, as well as with equally perfect competition. Note, therefore, that the burden of the following argument is that the labor market was comparatively imperfect.

⁸⁰ Cf. Zevin (unpublished work), p. 15; Lebergott, “Wage Trends, 1800–1900,” pp. 450–52, 454.

⁸¹ If homogenous labor, L′, is paid less than its marginal value product (supposing the firm to be a price-taker in the product market, though a monopsonist in the labor market), we have v > 0 in

where w is the money wage rate, and P is the price of cloth. The share of labor in total costs is therefore

But as we know from (13) that

it follows immediately that . Now suppose that the firm treats the price of its purchased material (cotton), P_c, as a parameter, so that profit-maximization subject to (13) leads to P_c/P = (∂Y/∂C), and therefore to:

If the underlying production function is first-degree homogenous, which the results discussed in the previous section indicate was likely to have been true for Blackstone, the share of value added in total cost of production is

In conjunction with the result obtained from (ii) and (iii), (v) permits us to assert the proposition cited in the text:

⁸² While labor's share of gross value added averaged (s_L/(1−s_c) = ) 0.414 for the Blackstone Co. over the period 1823–60, the estimate of the relative elasticity of labor inputs obtained from R-V in Table 3 is

By contrast, the regression estimated for Blackstone on the basis of the cumulated output measure of experience, R-IV, was seen to yield

⁸³ To calculate the exploitation coefficient, V, as defined in the text, we can make use of equations (ii), (iii) and (v) in the preceding footnote and write the general expression

This can be rearranged as follows, noting the relationship between the ratio of structural input coefficients and the ratio of the reduced-form parameters:

which is readily solved for V = v/(∂Y/∂L′), thus,

Substituting the estimate of the relative labor elasticity coefficient derived from regression R-V: Blackstone, and the average 1823–60 labor share in gross value added, we find:

.

⁸⁴ Zevin (unpublished work), p. 15.

⁸⁵ According to the Blackstone records assembled by Zevin (unpublished work, Table II, p. 13) wages as a proportion of gross value added averaged 0.489 in the years 1823–30, and .404 during 1830–60—with little secular change taking place within the latter interval. Thus, using equation (viii) of the previous footnote, we calculate V for 1823–30 as (1 − .489/.592) = .174, and V = (1 − .404/.592) = .318 for 1830–60.

⁸⁶ By employing the standard annuity formula for an asset of finite life, we may write:

where (I) is the real value of the investment in training one worker, (v) is the real value of the uniform annual repayment extracted by exploiting the worker at the rate V = v/(∂Y/∂L′), r is the annual opportunity cost rate of return required by the investing firm, and A is the period over which (I) must be amortized. The higher the likelihood of losing a trained worker, for any cause, the shorter the amortization period on which the training firm will insist.

Now the above expression may be divided through by (∂Y/∂L′), and solved for (I) in terms of r,A,V—which last has already been estimated at V = .301 for the period 1823–60—and (∂Y/∂L′) itself. Assuming the values r = .10 and A = 4, one finds

the relationship referred to in the text. It should be emphasized that this calculation is meant as nothing more than a check on the numerical plausibility of the estimated rate of exploitation implied by regression R-V:Blackstone.

⁸⁷ Cf. footnote 72, for comparison of the group and single firm estimates of in equations R-V.

⁸⁸ Notice that as the suppressed exponent α_c is the same in R-IV and R-V for the Blackstone Co., the ratio of the reduced-form parameter estimates will reflect the relative magnitudes of the underlying learning coefficient in the production function (13). I.e.,

where the subscript on the estimated overall learning coefficient refers to the argument of the learning function.

⁸⁹ The significance tests referred to are t-tests of the form

since in the two regressions from which the estimates are drawn the number of degrees of freedom (N) is identical. Testing R-III vs R-II for the sample group, we have the difference , which is highly significant: t = 7.24, with df pooled ≃32. Alternatively, we can compare the difference between the estimates of λ_T and λ_Q provided by the constrained version of the regression model:

which yields a still more strongly significant t-statistic, t = 12.55.

⁹⁰ From Appendix Table I it can be calculated, for example, that the annual growth rate of Q exceeded that of T (for the six-firm group) by as much as 10 percentage points in the latter half of the 1830's (1836–39), by 3.2 percentage points on average during the most of the next decade (1843–45), and by 2.2 percentage points on average—with little annual variation—throughout the remainder of the period (1850–60). It might be pointed out in this connection that the growth rate of Q for the six-firm group on the eve of the Civil War was 5.5 percent per year, not a spectacularly high figure. Indeed, it was probably no more rapid than the rate of growth of cumulated output in the New England cotton textile industry as a whole c.1858–60, and may well have been somewhat slower. The latter conclusions can be inferred from Davis and Stettler's (“The New England Textile Industry,” Table 4, p. 221) partial reconstruction of the historical record of textile production in New England.

⁹¹ See discussion of the findings of other studies, above, fns. 29 and 31, and note that the empirical range cited is established by those inquiries which used Q-measures of experience.

⁹² Comparing (13′) with (13), it is apparent that λ* in the latter would be equivalent to: . From (14b), however, one would find that given the condition

⁹³ Cf. n. 36, above. Actually the ratio of the value of materials to the value of shipments reported for the cotton textile industry by the Eighth Census was .51. But this gross materials share figure clearly overstates the (net) materials share that is appropriate in the present context. The reason it does so is simply that the gross value of materials purchased in the industry reflects not only (say) raw cotton purchases by spinning mills and integrated establishments, but purchases of yarn by weavers. It is not difficult to see that in general the industry's gross materials share will exceed the industry net materials snare, which will also be characteristic of integrated establishments like those under examination here.

Suppose that Spinners buy cotton worth cX_s and ship yam worth X_s to Weavers. The latter pay X_s = yX_w for it, working it up into X_w of cloth. The value input coefficient for yarn is y < 1, while that for cotton is cy < 1. Integrated establishments ship X_I worth of cloth, for which they will require cy X_I worth of cotton, assuming the unit prices and technical coefficients are the same for each operation, whether it is conducted by integrated or by non-integrated establishments. Thus, the industry flows are as follows:

From this it is readily seen that the ratio of the industry's gross materials purchases to its gross shipments will be:

But the (net) ratio of materials purchased from establishments outside the industry (cyX_w + cyX_I) to shipments of output destined for use outside the industry (X_w + X_I) is cy, identically equal to the materials share prevailing in the integrated plants alone.

The foregoing general proposition is confirmed by McGouldrick's (New England Textiles, pp. 144–45) fragmentary estimates of the share of the unit value of cloth output accounted for by the cost of raw cotton in 1839 and 1884. The estimates in question are derived from cloth and cotton price quotations, and technical coefficients—specifically, the weight of the average yard of standard cloth, and the net weight loss in the conversion of a pound of raw cotton into cloth—obtained from the accounts of companies operating integrated mills in New England. According to these figures, we find that in 1839 outlays for all purchased materials were roughly 0.35 of total cloth production costs, with the share of cotton costs alone accounting for 0.31. The corresponding shares for 1884—assuming fixed technical coefficients—would have been 0.41 for all purchased materials, and 0.37 for cotton taken alone. Compare these with the aggregate materials share (0.50) based on the U.S. Census of 1860.

⁹⁴ Specifically, the group learning coefficients range from λ* = 0.163, implied by the estimate of λ from R-Vc, to =.192, implied by R-III. Only one estimate is available for the Blackstone learning coefficient, that provided by Zevin's equation R-V, shown in Table 3.

⁹⁵ Recall that this interpretation can be justified on the basis of the production model represented in (13)—noting that the effect of learning on the rate of growth of the productivity of combined labor and capital inputs would, under the condition of constant returns to scale, be given by . Alternatively, the statement would follow directly from the form of the production specified in (13′), since λ′ = λ where the estimates of the latter are obtained as the coefficient of ln(T) in the regressions of Table 3. For the comparison of the Blackstone learning coefficient based on T with that estimated for the group, the estimates provided by fitting the constrained form of the regression—i.e., R-V (rather than R-III)—are used, since Zevin presents only an estimate of that kind for Blackstone. Note that if λ = .735, as in R-V: Blackstone, then the doubling of T between year 0 and year t will result in a 66.7 percent rise in the productivity of labor and capital, or a 40 percent fall in the level of unit primary factor costs between the two dates. See, above, fn. 31, for derivation of the relationship employed in making the computation.

⁹⁶ Cf. Table 1, above, and footnote 61; Zevin (unpublished work), p. 11.

⁹⁷ To avoid unnecessary complications, this assumes there were no significant interfirm transactions among the members of the group; and further, that the output, being homogeneous, was marketed at a uniform price.

⁹⁸ Proposition: For it is sufficient that (a) , and (b) U > 0, in the definitional expression

Proof: If (a) and (b) hold, we can write

and also

which, upon division by ∑T₁, and some rearrangement, becomes

.

We may now show, from the definition of U, that under conditions (a) and (b) the L.H.S. of (d) is identical to ∑θ_i/n. First, recalling the definition of θ_i given by equation (20), and invoking condition (b) we have:

Then, by summation over the index i, and division by n we find:

.

⁹⁹ Cf., e.g., Landes, D. S., The Unbound Prometheus (London: Cambridge University Press, 1969), pp. 41–42,Google Scholar 64–66, 80–88 (on the industry in Britain), and pp. 158–69 (in Belgium, Holland, France, Germany, and Switzerland); Lyaschenko, P. I., History of the National Economy of Russia to the 1917 Revolution (New York: Macmillan, 1949), pp. 333–39,Google Scholar on the cotton textile mills of the Moscow region in period 1799–1861. On die Japanese experience, notably the subsidization of pilot enterprises by the Meiji State, cf., Smith, T. C., Political Change and Industrial Development in Japan: Government Enterprise, 1868–1880 (Stanford: Stanford University Press, 1955), pp. 11,Google Scholar 26–27, 60–63; Y. Horie, “Modern Entrepreneurship in Meiji Japan” (esp., pp. 183–86); and J. Hirschmeier, “Shibusawa Eiichi: Industrial Pioneer” (esp., pp. 225–29), in The State and Economic Enterprise in Japan, Lockwood, W. W. (ed.), (Princeton: Princeton University Press, 1965)Google Scholar.

¹⁰⁰ The rationale for the calculation of Hicks-neutral efficiency growth is summarized in the definitions accompanying equation (2), above.

¹⁰¹ Implicitly we are asking the production function (13), as estimated by R-Vc for 1834–60, to fit the observations exactly for each of the two brief periods selected in Table 6: 1833–39 and 1855–59. But, since the regression model is based on a stochastic reformulation of the production relationships, there is no reason to expect the fit to be exact. The last line of Table 6 shows the small fractions—about onetwentieth and one-tenth, respectively—of the 1833–39 and 1855–59 labor productivity growth rates which remain unaccounted for by regression equation R-Vc. This statistically unexplained portion, however, is allowed to reflect itself in the table as deriving from “Exogenous Sources.”

¹⁰² Making use of the notation employed in Table 6, note, first, the definitions:

and

But, since we have argued for the validity of substituting Ẏ/Y = C݁/C in (i), it directly follows that the relative contribution of learning to total productivity growth is

and thus free of the influence of the value assumed for . It may be left to the reader to prove to his own satisfaction that the following magnitudes are also quite independent of the parameter , and therefore can be determined simply on the basis of the regression coefficients estimated in R-V^c:

¹⁰³ As is easily found from the preceding footnote, the magnitude Δ(λ*[Ṫ/T])/ Δ(A݁/A) = (.0148/.0143) is also independent of the value assigned to the parameter , so long as the underlying production function was such that the latter remained constant over the period of the change (Δ)—from 1833–39 to 1855–59.

¹⁰⁴ Note that this fraction, readily shown to be independent of the choice of an estimate for .

¹⁰⁵ Hence the rates computed for β_s exceed even the trend rates of capital augmentation which Table 4 would suggest as being consistent with regression model R-V (in which Ξ = 0) under the assumption that β_L = .010.

¹⁰⁶ For a n exposition of this view of the U.S. textile machinery industry, of. Strassman, W. Paul, Risk and Technological Innovation (Ithaca, N.Y.: Cornell University Press, 1959), pp. 76–101Google Scholar.

¹⁰⁷ To illustrate the point we may calculate the effects of a (large) 25 percent overstatement of . Assuming that rather than 0.5, for 1833–39 one obtains: A݁/A = [.0667 − (6/5)(.0074) − (4/5)(.0333)] = .0312, instead of .0260, and

On the same basis, for 1855–59 one obtains:

instead of .0117, and

.

Note that all that is involved is a change of the levels of the growth rates; the rates of retardation of A݁/A and of the portion due to learning effects remain unaffected by the recomputation, as does the relationship between the total productivity growth rate and its endogenous component.

¹⁰⁸ This comparison is based on the findings of a forthcoming study of post-World War II U.S. manufacturing efficiency, which Moses Abramovitz and I have prepared in connection with the Stanford SSRC Study of Economic Growth in the United States. Note that whereas the estimated productivity growth rates in Table 6 refer essentially to best-practice productivity within the cotton textiles branch, the efficiency growth rates cited for two-digit manufacturing industries pertain to averagepractice productivity, and to much wider industrial segments. The comparison is favorable to the sample group of ante-bellum cotton textile companies on the latter count, but is stacked against them on the former—so long as it is possible in the short-run to raise average productivity rapidly by closing the gap between averagepractice and best-practice.

¹⁰⁹ Cf. Appendix Table I for underlying data from which the annual rates of change in the experience-index T, and hence, the learning effects—0.163 (ΔT/T)—have been computed. The learning coefficient is naturally the same one used in Table 6.