¹ Ricardo, David, Principles of Political Economy and Taxation, The Works and Correspondence of David Ricardo, vol. 1, ed. Sraffa, Piero (Cambridge, 1951),Google Scholar chap. 31, “On Machinery,” p. 395.Google Scholar

² Economic historians (and theorists) have an unfortunate habit of casually interchanging the terms machinery and capital, in spite of the fact that machinery rarely constitutes more than one sixth of a nation's stock of reproducible tangible assets. The result is frequent confusion regarding whether statements about machinery apply to the capital stock as a whole or only to a small portion of it. For contemporary examples of the use of the term machines or machinery, apparently as representative of capital in general, see Temin, Peter, “Labor Scarcity and the Problem of American Industrial Efficiency in the 1850's”, this JOURNAL, 26 (09 1966), 296;Google ScholarTemin, Peter, “General Equilibrium Models in Economic History”, this JOURNAL, 31 (03 1971), 63;Google Scholar or Rosenberg, Nathan, “Economic Development and the Transfer of Technology,” Technology and Culture, 11 (1970)CrossRefGoogle Scholar, reprinted in Rosenberg, Nathan, Perspectives on Technology (Cambridge, 1976), p. 159.CrossRefGoogle Scholar On the breakdown of national capital stocks, see Field, Alexander J., “On the Unimportance of Machinery” (January 1983), unpublished.Google Scholar

³ Elsewhere Ricardo wrote “…old countries are constantly impelled to employ machinery, and new countries labour” (Principles, chap. 1, section 5, footnote, p. 41). Ricardo claimed that the “value” of wages was lower in the United States, that is, the embodied labor content of wage goods—primarily food—was lower. He rejected Adam Smith's convention of measuring the value of money wages by the commodities they could command, although it is Smith's convention that has been most widely accepted by modern economists. His statement that land scarcity “raises labour” (Principles, p. 395) does not necessarily imply that British workers' consumption bundles were strictly superior to those in America, and the argument of this paper is not dependent on such a claim.Google Scholar For a detailed study of the composition of family budgets in Great Britain, France, and the United States, see Fishlow, Albert, “Comparative Consumption Patterns, the Extent of the Market, and Alternative Development Strategies,” in Micro Aspects of Development, ed. Ayal, Eliezer B. (New York, 1973), pp. 41–79.Google Scholar

⁴ Rothbarth, Erwin, “Causes of the Superior Efficiency of U.S.A. Industry Compared with British Industry,” Economic Journal, 56 (1946), 383–90;CrossRefGoogle ScholarHabakkuk, H. J., American and British Technology in the Nineteenth Century: The Search for Labour-Saving Inventions (Cambridge, 1962). As a matter of record, the Rothbarth-Habakkuk treatment combined this argument with a wide variety of others, perhaps the most important of which were 1) that factor scarcity “biased” (the search for new technologies towards a particular (labor-saving) end of the technological spectrum, and 2) that Americans were more technologically optimistic about the future availabilities of new cheaper production options than were the British entrepreneurs. Relatively little attention is given to either of these arguments in this paper. The theoretical logic of the first argument is flawed in that entrepreneurs ought to prefer and adopt cost-reducing techniques regardless of their factor-saving character. In regard to the second argument, I see no compelling reason to believe that American entrepreneurs were more technologically optimistic than their British counterparts.Google Scholar

⁵ Such differences may also result from differences in rules influencing the disposition of national income, although the range over which this latter influence can operate is restricted by the subsistence requirements of workers. For example, the successful introduction or reintroduction of slavery or coerced labor in a region might increase the profit rate—assuming unchanged labor effort. Alternately, the absence of unions might imply a lower labor share, and thereby a higher profit share and rate. In both cases, however, there is a natural limit to how high such institutional variation could push interest/profit rates, absent new technological availabilities. The evidence of Section III indicates that the bargaining power of American workers, and therefore labor's share, was not sufficiently high to counteract the favorable influence of land abundance on interest/profit rates. On the relationship between markets and legal or conventional rules, see Field, Alexander J., “Microeconomics, Norms and Rationality,” Economic Development and Cultural Change, forthcoming.Google Scholar

⁶ Temin, Peter, “Labor Scarcity and the Problem of American Industrial Efficiency in the 1850's”;Google ScholarFogel, Robert, “The Specification Problem in Economic History”, this JOURNAL, 27 (09 1967), 283–308;Google ScholarJones, Ronald W., “A Three Factor Model in Theory, Trade and History,” in Trade, Balance of Payments and Growth—Papers in International Economics in Honor of Charles P. Kindleberger, ed. Bhagwati, Jagdish N., Jones, Ronald W., Vanek, Jaroslav, Mundell, Robert A. (Amsterdam, 1971), pp. 3–21.Google Scholar

⁷ Some authors appear to believe that the introduction of land or natural resources into the manufacturing production function resolves the difficulties associated with, for example, Temin's formalization of the Rothbarth-Habakkuk argument. To the extent one is interested in the effect of land abundance on aggregate capital intensity, this claim is false in principle, because by assumption in specific factor models, aggregate K and aggregate L are fixed exogenously, regardless of the number of sectors or which inputs enter which sectors. In practice such models have been used to rationalize higher American manufacturing capital intensity. Yet, as Section II argues, it is the reverse that needs to be explained, in manufacturing and elsewhere in the economy. See Fogel, , “The Specification Problem”;Google ScholarAmes, Edward and Rosenberg, Nathan, “The Enfield Arsenal in Theory and History,” Economic Journal, 78 (12 1968), 827–42. If one abandons the fixed (inelastic) supplies of inputs assumptions in these models, permitting factor supplies to vary in response to their price, one can, by assumption, produce any desired effect of the addition of land on the aggregate capital/labor ratio. But aside from the obvious fact that the direction of such an effect is not easily deduced on a priori grounds, there remains a limitation in using such a modified model. These models can only characterize technologies by the proportion in which physical factor inputs are combined. They cannot therefore easily distinguish between technologies that use physical factors in similar proportions, but require larger or smaller stocks of them, and depreciate them, respectively, at lower or higher annual rates.CrossRefGoogle Scholar

⁸ Habakkuk, , American and British Technology, pp. 56–57.Google Scholar

⁹ Ibid., pp. 60–61.

¹⁰ It is difficult for the twentieth, let alone the nineteenth, century to maintain that individual entrepreneurs actually cost out correctly all possible alternatives in their production decisions. In the aggregate and over time, however, one might still believe that “natural selection” in the market place would favor the survival of those firms whose rules of thumb or standard operating procedures were advantageous given the factor endowments of a region. This idea is developed in Friedman's, Milton “The Methodology of Positive Economics,” in Essays in Positive Economics (Chicago, 1953), especially p. 22.Google Scholar For a recent treatment, see Nelson, Richard R. and Winter, Sidney G., An Evolutionary Theory of Economic Change (Cambridge, Massachusetts, 1982).Google Scholar

¹¹ Habakkuk, , American and British Technology, p. 86.Google Scholar

¹² Ibid., p. 88. The citation is from Watkin, E. W., A Trip to the United States and Canada (1851), p. 124.Google Scholar See also Fishlow, Albert, who writes: “… gestation periods were foreshortened by the practice of opening small sections of line as they were completed. Engineering considerations always ranked second to early completion, as the small extent of tunneling and double tracking, among other characteristics, bears witness… [this] considerably eased access to funds in a fragmented and imperfect capital market… American technological ingenuity found ways to economize on the absolute quantities of capital needed. Durability was sacrificed for lower capital costs…” American Railroads and the Transformation of the Antebellum Economy (Cambridge, Massachusetts, 1965), p. 308.Google Scholar

¹³ Quoted in Nevins, Allan, Abram S. Hewitt (New York, 1935), p. 246.Google Scholar Cited in Temin, Peter, “Labor Scarcity and the Problem of American Industrial Efficiency”, p. 291Google Scholar

¹⁴ Potter, M. R., “The South Chicago Works of the North Chicago Rolling Mill Company,” Iron and Steel Institute Journal, 30 (1887), 163–79Google Scholar, cited in Berck, Peter, “Hard Driving and Efficiency: Iron Production in 1890,” this JOURNAL, 38 (12 1978), p. 880.Google Scholar

¹⁵ Berck, Peter, “Hard Driving”, p. 894.Google Scholar

¹⁶ Floud, Roderick, The British Machine Tool Industry, 1850–1914 (Cambridge, 1976), p. 102.CrossRefGoogle Scholar

¹⁷ Ibid.

¹⁸ Olmstead also argues that David's use of a 6 percent discount rate understand the true cost of funds. Olmstead, Alan, “The Mechanization of Reaping and Mowing in American Agriculture, 1833–1870,” this JOURNAL, 35 (06 1975), 327–53, pp. 331–34;Google ScholarDavid, Paul A., “The Mechanization of Reaping and Mowing in the Antebellum Midwest,” in Industrialization in Two Systems: Essays in Honor of Alexander Gerschenkron, ed. Rosovsky, Henry (New York, 1966), pp. 3–39.Google Scholar

¹⁹ Bogue, Allan, From Prairie to Corn Belt (Chicago, 1963), p. 145.Google Scholar

²⁰ It is true that England did not face the same need for inland shallow draft water vessels. What is remarkable, however, is the similarity of the useful lives of these vessels with those reported for other elements of the American capital stock. Chevalier, Michel, Letter sur l'Amérique du Nord, 2 vols. (Paris, 1836) vol. 2, letter 21, p. 14. I am indebted to Charles Kindleberger for drawing my attention to these particular sections of Chevalier's work. Many of Chevalier's remarks on economic matters are unavailable in the excerpted English edition of this work, entitled Society, Manners and Politics in the U.S.A.Google Scholar

²¹ Ibid., vol. 2, letter 21, p. 25. This American concern with the value of time in production seems to have spilled over into other aspects of daily life. The widespread acceptance of Benjamin Franklin's maxim that time is money is nicely illustrated in Chevalier's description of American dining habits: “Dans les hotels et sur bateaux à vapeur, lorsque l'heure des repas approche, la porte de la salle à manger est assiégée. Dès que la cloche sonne, on se rue, et en moins de dix minutes, toutes les places sont envahies. Au bout d'un quart d'heure, sur trois cents personnes, deux cents sont sortie de table; dix minutes après, tout a disparu.” (, Chevalier, Lettres, vol. 2, endnote 19, p. 453.) To this day, Europeans remark disparagingly about the American attitude toward time. [Google ScholarScitovsky, Tibor, The Joyless Economy (Oxford, 1976)]. Although this attitude, to the extent that it now prevails, is probably deeply ingrained in American culture, its original source may have been in part the imperatives in production which a high interest rate economy produced.Google Scholar

²² Chevalier, Michel, Lettres, vol. 2, letter 21, p. 42.Google Scholar

²³ Chandler, Alfred, The Visible Hand: The Managerial Revolution in American Business (Cambridge, Massachusetts, 1977), p. 241.Google Scholar

²⁴ Rosenberg, , “Economic Development and the Transfer of Technologies,” in Perspectives on Technology (Cambridge, 1976), p. 161.CrossRefGoogle Scholar

²⁵ It is interesting to note that the very first example of a capital-saving invention given in Paul Samuelson's text is the “case of a cheap computer that enabled firms to get along with much less inventory.” Samuelson, , Economics, 11th ed. (New York, 1980), p. 692.Google Scholar

²⁶ Habakkuk and others have tried to explain the more rapid adoption of mass production and standardization in America by emphasizing the different supply schedules for skilled and unskilled labor in the two countries. Habakkuk argued that in the United States unskilled labor was scarce and in inelastic supply due to high earnings opportunities in agriculture; mechanization, he suggested, used skilled labor more intensively than unskilled (p. 21). But Rosenberg has pointed out that Habakkuk relied for his skill premia generalizations on Victor Clark's “unwarranted” interpretation of the Zachariah Allen data. [Rosenberg, Nathan, “Anglo-American Wage Differentials in the 1820's,” this JOURNAL, 27 (06 1967), 221–29. Zachariah Allen's data show the skill premium for “best machine makers,” but not for skilled labor, to be higher for Britain]. Donald Adams's work has also questioned whether the skill differential was higher in Great Britain.Google ScholarAdams, Donald R. Jr, “Some Evidence on English and American Wage Rates 1790–1830,” this JOURNAL, 30 (09 1970), 499–520. Moreover, previous research I have done has questioned whether American industrialization was, in its early stages, skilled labor intensive.Google ScholarField, Alexander J., “Skill Intensity in Early industrialization: The Case of Massachusetts,” Journal of Human Resources, 15 (Spring 1980), 149–75. Both components of the skilled/unskilled labor explanation of the more rapid adoption of mass production and standardization in the United States rest on uncertain empirical foundations.Google Scholar

²⁷ The potential significance of inventory savings is reflected in Robert Fogel's calculations of the social savings of American railways on freight account, which initially showed negative savings. Only when he took into account the benefit of reduced inventories associated with shorter transshipment periods did these savings turn mildly positive. Fogel, Robert, Railroads and American Economic Growth: Essays in Econometric History (Baltimore, 1964).Google Scholar See also Fishlow, , American Railroads, p. 24.Google Scholar

²⁸ Edwards, Richard, Contested Terrain: The Transformation of the Workplace in the Twentieth Century (New York, 1979), pp. 97–104.Google Scholar

²⁹ Levine, A. L., Industrial Retardation in Britain, 1880–1914 (New York, 1967), p. 60.Google Scholar

³⁰ Chevalier, , Letters, vol. 2, letter 21, p. 20.Google Scholar

³¹ British Board of Trade, Cost of Living in American Towns: “Report of an Enquiry by the Board of Trade into Working Class Rents, Housing and Retail Prices together with the Rates of Wages in Certain Occupations in Principal Industrial Towns of the United States of America. With an Introductory Memorandum and a Comparison of Conditions in the United States and the United Kingdom.” (London, 1911), c.d. 5609, pp. lx.Google Scholar

³² British Board of Trade, Cost of Living in American Towns, pp. lxxxiv–lxxxv, cited by Albert Fishlow, “Comparative Consumption Patterns,” p. 76.Google Scholar

³³ Goldsmith, Raymond, “The Growth of Reproducible Wealth in the United States” Income and Wealth of the United States, Trends and Structure (Cambridge, 1952), pp. 306–7Google Scholar, cited in Davis, Lance, Easterlin, Richard, and Parker, William et al., American Economic History: An Economist's History of the United States (New York, 1972), pp. 308–9.Google Scholar

³⁴ Feinstein, Charles, “Capital Formation in Great Britain,” in Cambridge Economic History of Europe, VII, ed. Mathias, Peter and Postan, M. M. (Cambridge, 1978), p. 72. The 1840 United States Census reports a total population of 17,069,453. The population of Great Britain in the 1841 census was approximately 18.5 million. American population was, however, growing faster: 1861 British population is about equal to 1850 American population (23.2 million). By 1900 the United States population was just over twice that of Great Britain's (76 million versus 37 million).Google Scholar See Mitchell, B. R., Statistical Appendix to Cipolla, Carlo, ed., The Fontana Economic History of Europe, vol. 4, part 2 (London, 1973), p. 747.Google Scholar

³⁵ Behavior with respect to these two types of assets is very similar. By economizing on inventory stocks one reduces the average length of time a component of the stock is held, and in that sense, its “durability,” or persistence in the capital stock, is lower.Google Scholar

³⁶ Goldsmith, , “Growth of Reproducible Wealth,” pp. 306–7.Google Scholar

³⁷ Giffen, Robert, The Growth of Capital (London, 1889), p. 119.Google Scholar

³⁸ Feinstein, , “Capital Formation in Great Britain,” pp. 40, 70, 72. Feinstein's railway estimate, based on Mitchell, is reduced 14 percent to take account of railway investment representing land purchases.Google Scholar See Mitchell, B. R., “The Coming of the Railway and United Kingdom Economic Growth,” this JOURNAL, 24 (09 1964), appendix.Google Scholar See also Field, Alexander J., “On the Unimportance of Machinery,” Table 5.Google Scholar

³⁹ Angus Maddison shows British GDP per capita exceeding U.S. by 13 percent in 1820 and 18 percent in 1870. “A Comparison of Levels of GDP Per Capita in Developed and Developing Countries, 1700–1980,” this JOURNAL, 43 (03 1983), 27–41, p. 30.Google Scholar

⁴⁰ Homer, Sidney, A History of Interest Rates, (New Brunswick, 1963), p. 292.Google Scholar

⁴¹ “History tends to overemphasize newsworthy defaults and leaves in obscurity periodic debt repayments between nations.” Homer, , Interest Rates, p. 202.Google Scholar

⁴² Nineteenth-century Consolidated Canadian, British Columbian, New Zealand, and New South Wales sterling bonds issued in London commanded premia above British consol yields in the range of 200 basis points, premia comparable to those commanded by United States government securities. These interest rates must provide a lower bound for the terms on which financial capital was available to entrepreneurs in these regions, since if local lenders were willing to provide funds at lower rates to local entrepreneurs, they would presumably have been willing to do so on the securities of their governments, which had access to the taxing power if necessary to insure repayment. See Homer, , Interest Rates, p. 203.Google Scholar

⁴³ Edelstein, Michael, “Rates of Return on U. K. Home and Overseas Portfolio Investment in the Age of High Imperialism,” Explorations in Economic History, 13 (07 1976), 283–329.CrossRefGoogle Scholar

⁴⁴ Malkiel, Burton, “Risk and Return: A New Look,” National Bureau of Economic Research Working Paper No. 700 (June 1981).Google Scholar

⁴⁵ Grossman, S.J. and Stiglitz, J. E., “The Impossibility of Informationally Efficient Markets”, American Economic Review, 70 (06 1980), 393–408, p. 404.Google Scholar

⁴⁶ Sharpe, William F., Portfolio Theory and Capital Markets (New York, 1970), especially chap. 4.Google Scholar

⁴⁷ Although American per capita incomes were in the same range as the British, the American population grew faster (see footnote 34). As a result, saving rates were applied to a considerably larger income base in the United States over time, resulting in a more rapidly growing volume of American saving. This was compounded by a marked rise in the United States saving rate over the course of the nineteenth century. According to the assumptions of this model, these developments should have contributed to a capital outflow from the United States to Britain. Paul David estimates that the flow of saving available for net additions to reproducible tangible assets increased from 11.8 percent of real gross output in 1800–1835 to 15.9 percent in 1890–1905. David, , “Invention and Accumulation in America's Economic Growth: A Nineteenth Century Parable,” in International Organization, National Policies and Economic Development, ed. Brunner, Karl and Meltzer, Allan (Amsterdam, 1977), pp. 179–227. Deane and Cole's data for 1860 through 1913 indicate that the sum of gross domestic fixed capital formation and net foreign investment as a percent of GNP at market prices peaked in 1872 at 14.9 percent. In 1860 that figure stood at 8.5 percent; in 1900 at 11.3 percent. Since the American estimates are of net savings, it is possible that American saving rates were higher by the turn of the twentieth century than British rates.Google ScholarDeane, Phyllis and Cole, W. A., British Economic Growth, 1688–1959 (Cambridge, 1969), Appendix Ill, Table 91, p. 337.Google Scholar

⁴⁸ Clarke, Richard N. and Summers, Lawrence H., “The Labour Scarcity Controversy Reconsidered,” Economic Journal, 90 (03 1980), 129–39, p. 135.CrossRefGoogle Scholar

⁴⁹ A partially analogous example can be found in the secondary market for United States Treasury Bonds. Long-term Treasury issues with low coupons have significantly smaller yields to maturity than do more recently issued securities with higher coupons. The reason is that high income earners prefer to take their yield as capital gains rather than interest. Their desire for this is so strong that a risk-adjusted yield differential is created with no inherent incentives for arbitrage. In principle, portfolio covariance considerations could operate in an analogous fashion.Google Scholar

⁵⁰ Habakkuk, , American and British Technology, p. 27: “The opportunity costs of industrial finance were clearly higher in the U.S.A. as a whole than in Britain.” Habakkuk goes on to imply that correcting for higher American risk would diminish but not eliminate this differential. See also pp. 69–70, 90. Peter Temin, “Labor Scarcity and the Problem of American Industrial Efficiency in the 1850's,” p. 291: “The interest rate was consistently higher than the rate in Britain during the first half of the nineteenth century.” Temin's generalization was based on data presented in Homer, A History of interest Rates. A recent treatment of American financial intermediation concluded that “Despite the potential profitability of investment … capital was not being efficiently mobilized across regional or industrial borders through at least the first four-fifths of the nineteenth century.” If this were true, one would expect East-Midwest regional differences in technology and organization in the United States itself to have replicated, on a smaller scale, some of the characteristics noted in the aggregate United States-British comparison. Lance Davis, Richard Easterlin, William Parker, et al., American Economic History, p. 328.Google Scholar

⁵¹ See Friedman, Milton and Schwartz, Anna J., Monetary Trends in the United States and the United Kingdom: Their Relation to Income, Prices and Interest Rates: 1867–1975 (National Bureau of Economic Research, 1982). For example, “In 1876 the U.S. was a debtor and borrower country: the U.K. a creditor and lending country” (pp. 5–14, manuscript).Google Scholar Their data is based largely on Macaulay, F. R., The Movements of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856 (New York, 1938), pp. A145–A161.Google Scholar

⁵² Bateman, Fred and Atack, Jeremy, “The Profitability of American Agriculture in 1860,” in Research in Economic History, ed. Uselding, Paul, 4 (1979), 87–125, especially p. 116.Google Scholar See also Vedder, Richard K. and Gallaway, Lowell E., “The Profitability of Antebellum Manufacturing: Some New Estimates,” Business History Review, 54 (Spring 1980), 92–103.Google Scholar

⁵³ For example, Schefold, Bertram, “Different Forms of Technical Progress,” Economic Journal, 86 (12 1976), 806–19, p. 807, footnote 1.CrossRefGoogle Scholar

⁵⁴ Roemer, John, “Continuing Controversy on the Falling Rate of Profit: Fixed Capital and Other Issues,” Cambridge Journal of Economics, (December 1979), 379–98.Google Scholar

⁵⁵ The joint product treatment formally eliminates (while substantively maintaining) the different depreciation rates associated with more or less durable fixed capital: all inputs are fully depreciated over the course of a production period, although a capital input of a certain age “produces” an older vintage capital good along with output. The von Neumann framework has not previously been used systematically to examine technique shifts involving the replacement of whole subsystems of techniques with larger subsystems involving longer durability (and fundamentally different) capital goods. There are problems with the formulation. For example, it implies, in violation of national income accounting conventions, that older vintage goods form part of the gross output of an economy: a convention that can lead to difficulties when calculating the impact of the adoption of capital-deepening technologies on capital-output ratios. In an unpublished paper I develop the fixed capital argument of this paper within the joint product framework, although for heuristic reasons I have chosen here to develop these ideas somewhat more simply. See Field, Alexander J., “Induced Technique Shifts and Capital Deepening in a Linear Model” (March 1983).Google Scholar

⁵⁶ These are the double switching (reswitching) and capital-reversing conclusions associated with the work of Sraffa, Piero, Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory (Cambridge, 1960).Google Scholar

⁵⁷ Depreciation flows per unit output are lower for B.2, but interest charges per unit output may be higher, depending upon the prevailing interest/profit rate. B. I and B.2 represent fundamentally different techniques, although each individually can be thought of as constant returns to scale.Google Scholar

⁵⁸ Demand variation, it is true, could influence the margin of cultivation, therefore the degree of superiority of A.2, and therefore the endogenous variables. This possibility is abstracted from.Google Scholar

⁵⁹ This conclusion follows both from the one-commodity model customarily associated with the 1815 Essay, and the multicommodity model associated with the Principles. The one-commodity model is relatively easy to understand. If grain is the only output of an agricultural economy, and is produced solely with grain input (into the mouths of workers and into the ground as seed) then the economy's profit rate on the marginal plot of land can be calculated as (output-input)/input, all quantities measured in grain. This rate would be regulated fundamentally by the productivity of the worst land under cultivation, and secondarily by the grain requirements or wages of workers. In the multicommodity Principles model, relative commodity prices would be determined (approximately) by the embodied labor in the commodity relative to the embodied labor in an ounce of gold. Rising embodied labor content, due to the use of worse land, would drive up the money price of grain (and thus wages). Other commodity prices would be unaffected in the absence of technical change and profits would be squeezed by constant output and rising wage costs. The money wage rate would therefore transmit the decline in the “grain” rate of profit to other sectors of the economy. See Ricardo, David, “An Essay on the Influence of a Low Price of Corn on the Profits of Stock,” in The Works and Correspondence of David Ricardo, ed. Sraffa, Piero (Cambridge, 1951), vol. 4, p. 13;Google ScholarRicardo, David, Principles, chap. 6, pp. 115, 123, chap. 31, p. 395.Google Scholar

⁶⁰ Consider an n × n augmented input matrix M giving the input requirements per unit output for each commodity in the system, including the input requirements channeled indirectly through labor. Workers consume a fixed commodity bundle; direct labor requirements per unit output determine (proportionally) these indirect commodity inputs. Assume moreover that M is indecomposable (that is, all commodities are basic, entering directly or indirectly into each other's production). Equilibrium prices p₀ and interest/profit rate r₀ will be given by

where p₀ is an n element column vector of equilibrium prices and r₀ is the equilibrium profit rate. Let e = the maximal eigenvalue of M, that is, the maximal root to the characteristic polynominal, calculated as the determinant of M's characteristic matrix. It can then be shown that r = (I - e)/e. Two theorems of Frobenius are relevant: first, only the maximal eigenvalue has associated with it nonnegative (meaningful) prices, and second, reduction of any element of M will lower its maximal eigenvalue. Thus, if a newly available technique can be represented as a new vector in M such that one (or more) elements (input requirements) are reduced, it will reduce e, and therefore raiser. What if the new technique is cost-reducing at initially prevailing relative prices and r, but lowers some elements of a vector and raises others? How can we be sure it will raise r? Take the new vector m_n, choose any element in it and raise it until per unit costs using initial prices and rare equivalent to costs using the original vector (m₀). r_h using this hypothetical vector will not differ from r₀. The new vector m_n differs from the hypothetical vector m_h only in that one element is lower. Therefore, r_n > r_h, and since r_h = r₀, r_n > r₀. See Bowles, Samuel, “Technical Change and the Profit Rate: A Simple Proof of the Okishio Theorem,” Cambridge Journal of Economics, 5 (06 1981), 18386;CrossRefGoogle ScholarOkishio, Nobuo, “Technical Change and the Rate of Profit,” Kobe University Economic Review, 7(1961), 85–99. Okishio acknowledged the Ricardian ancestry of the theorem, and Marx's (incorrect) objection to it (p. 91).Google Scholar

⁶¹ This switch point is obtained by dividing B.2 through by (1 + α). The left hand side of this expression now gives the cost of obtaining B using B.2. This expression is equated to the cost of obtaining B using B. 1, and the switch point (r where costs using either technique are the same) solved for.

⁶² Intuitively, r rises, even though sector B output is lower, because profits are paid on a smaller sectoral and aggregate capital stock. Formally, this is another instance of the Ricardo-Okishio theorem.

⁶³ This assumption is consistent with the rough equality of the value of 1860 British and 1850 American national income and output.

⁶⁴ If two goods each embodied 100 labor hours, but one took two years and the other one year to produce, the former would command a higher price, because capitalists would demand and receive compensation for not being able to take out and reinvest their capital after one year. Ricardo believed, however, that “the greatest effects which could be produced on the relative prices of these goods [by variation in interest/profit rates] could not exceed six or seven percent…”(Principles, chap. 1, section 4, p. 36)Google Scholar, leading George Stigler to write an article entitled “Ricardo and the 93% Labor Theory of Value,” American Economic Review, 48 (06 1958), 357–67.Google Scholar