¹ The idea that increasing returns existed or that there were “advantages of large scale production” can be found in a number of studies: Whitten, David O., “Antebellum Sugar and Rice Plantations, Louisiana and South Carolina: A Profitability Study,” Ph.D. dissertation, Tulane University, 1970, p. 36Google Scholar; Gates, Paul W., The Economic History of the United States, vol. iii, The Farmers' Age: Agriculture, 1815–1860 (New York: Holt, Rinehart, and Winston, 1960), p. 127Google Scholar; Swan, Dale Evans, The Structure of the Rice Economy: 1859 (Ph.D. dissertation, University of North Carolina, 1972; Arno Press, 1975), p. 90Google Scholar. Swan, on the basis of similarities between the sugar and rice sectors, suggested that his findings of scale economies in rice might be easily implied to apply to sugar.

² Fogel, Robert William and Engerman, Stanley L., Time on the Cross: The Economics of American Negro Slavery (Boston: Little, Brown and Company, 1974), Evidence and Methods, pp. 138–45Google Scholar. Other authors, however, have come to the opposite conclusion. This paper suggests misspecification of their models may explain their failure to identify increasing returns. See Vedder, Richard, Klingaman, David, and Galloway, Lowell, “The Profitability of Antebellum Agriculture in the Cotton Belt: Some New Evidence,” Atlantic Economic Journal, 2 (Nov. 1974) pp. 30–47CrossRefGoogle Scholar; and Vedder, Richard and Stockdale, David, “The Profitability of Slavery Revisited: A Different Approach,” Agricultural History, 49 (Apr. 1975), pp. 392–404Google Scholar.

³ Schmitz, Mark, “Post-Bellum Developments in the Louisiana Cane Sugar Industry,” Business and Economics History, 5 (1976), pp. 88–101Google Scholar.

⁴ The bulk of the above information comes from annual reports on Louisiana's sugar planters by Champomier, P. A.: Statement of the Sugar Crop of Louisiana, 1849/50–1859/60 (New Orleans: 1850–1861)Google Scholar. This invaluable report provided output, locational, and capital information for each plantation as well as aggregate data on the sector. For further discussion of refining before and after the war see Schmitz, “Post-Bellum Developments,” pp. 92–98.

⁵ Prichard, Walter, “Routine on a Louisiana Sugar Plantation Under the Slavery Regime,” Mississippi Valley Historical Review, 14 (Sept. 1927), p. 177CrossRefGoogle Scholar.

⁶ Sitterson, J. Carlyle, “Financing and Marketing the Sugar Crop of the Old South,” Journal of Southern History, 10 (1944), p. 197Google Scholar.

⁷ My definition of scale is limited to efficiency changes across size. An alternative definition would be to include anything that reduced average cost but I do not reach this for and, in feet, argue that other factors that would affect the cost function, namely factor prices, are not important for the sugar sector.

⁸ An additional problem that I do not consider here is that Y must be an aggregate of many crops and products. Since I was unable to isolate the inputs that went only to sugar I was unable to bypass this problem.

⁹ Zellner, A., Kmenta, J., and Dreze, J., “Specification and Estimation of Cobb-Douglas Production Function Models,” Econometrica, 34 (Oct. 1966), pp. 784–95CrossRefGoogle Scholar. See also Hodges, Dorothy J., “A Note on the Estimation of Cobb-Douglas and CES Production Function Models,” Econometrica, 37 (Dec. 1969), pp. 721–25CrossRefGoogle Scholar for an extension of ZKD to the CES function. The importance of the ZKD criticism and their alternative model lies in the properties of the estimators. The ZKD approach does lead to best linear unbiased estimators (BLUE) under the assumptions specified below. That these assumptions have been left unstated should be bothersome to those concerned with specification errors in historical research. More importantly, the ZKD conclusion that production functions can be efficiently estimated in the absence of cost and wage data does not seem to have found a wide audience among economic historians. This discussion, therefore, is intended to restate some basic propositions that have been accepted and utilized outside this field.

¹⁰ Zellner et al., “Specification,” p. 326.

¹¹ Ibid., p. 327.

¹² The construction of the sugar samples is discussed at length in Chapter 3 of my dissertation: Economic Analysis of Antebellum Sugar Plantations in Louisiana (University of North Carolina, 1974; Arno Press, 1977)Google Scholar. A shortened version, of this chapter serves as an appendix to this paper. Copies are available upon request.

¹³ Hutchinson, William K. and Williamson, Samuel H., “The Self-Sufficiency of the Antebellum South: Estimates of Food Supply,” Journal of Economic History, 31 (Sept. 1971), pp. 596–98CrossRefGoogle Scholar.

¹⁴ Schmitz, Mark, “Farm Interdependence in the Antebellum Sugar Sector,” Agricultural History, (Oct. 1977)Google Scholar.

¹⁵ Gallman, Robert E., “Self-Sufficiency in the Cotton Economy of the Antebellum South,” Agricultural History, 44 (Jan. 1970), 5–24Google Scholar; Raymond C. Battalio and John Kagel, “The Structure of Antebellum Southern Agriculture: South Carolina, A Case Study,” ibid., 25–38.

¹⁶ The price data came mainly from Cole, Arthur H., Wholesale Commodity Prices in the United States, 1700–1861, Statistical Supplement (Cambridge: Harvard University Press, 1938)Google Scholar and various dates of “Wholesale Prices” published in the New Orleans Bee. Additional data and quotations for minor crops were found in various wholesale price columns, including those in the Times Picayune, American Agriculturist, and Louisville Courier.

¹⁷ Fogel and Engerman, Time on the Cross, pp. 72–82. Further analysis of prices supplied by Fogel and Engerman also supported the weighting system.

¹⁸ Battalio and Kagel, “Antebellum Southern Agriculture,” p. 27; Foust, James D. and Swan, Dale E., “Productivity and Profitability of Antebellum Slave Labor: A Micro-Approach,” Agricultural History, 44 (Jan. 1970), pp. 42–43Google Scholar.

¹⁹ A problem that I cannot handle is the role of hired labor on the farms. The possible impact of its exclusion is discussed below.

²⁰ Wright, Gavin, “Economic Democracy and the Concentration of Agricultural Wealth in the Cotton South, 1850–1860,” Agricultural History, 44 (Jan. 1970), pp. 70–79Google Scholar; Menn, Joseph Karl, The Large Slaveholders in Louisiana—1860 (New Orleans: Pelican Publishing Company, 1961), pp. 11–14Google Scholar.

²¹ Variation in the estimated value of improved land was found, but it was due mainly to inaccuracy in the reported farm values. For example, the average value per improved acre ranged from $25 to $425 for the fifteen largest farms in the 1860 sample. These inaccuracies were an additional reason for preferring the acreage input. The rejection of an increasing value for sugar farms did not mean that the same was true for all farms in the region. Across all farms the value per acre did rise, and sugar farms had higher values than non-sugar farms.

²² U.S. Bureau of the Census, Tenth Census of the United States 1880, vol. V, Report on Cotton, part 1 (Washington: GPO, 1884), pp. 3, 4, 110–95Google Scholar; Lockert, Samuel H., Second Annual Report of the Topographical Survey of Louisiana (1870) (New Orleans, n.p., 1871)Google Scholar; Rapley, E. E., The Soils and Products of Southwestern Louisiana, USDA, Office of the Secretary, Report no. 35 (Washington: GPO, 1884)Google Scholar.

²³ Aiena, Joseph, Plantations on the Mississippi from Natchez to New Orleans (New York: Rand-McNally, 1931)Google Scholar. This map is a copy of the original by Persac. Each farm in the river parishes is drawn to scale in this map and the owners' names are noted. I would like to thank the LSU Library for making this map available.

²⁴ Champomier, Sugar Crop, 1859–60, pp. 6, 10.

²⁵ Lockett, Topographical Survey, p. 9.

²⁶ Alternative specifications of the model are available. A generalized two-input CES function (using a capital input aggregated from land and machinery) was estimated with the results being inconclusive as to the value of elasticity of substitution (see Mark Schmitz and Donald Schaefer, “Scale and Substitution Reconsidered,” manuscript, 1977). Since the scale parameter is unaffected by t h e substitution specification and I wished to use machinery as a separate input, the function was not used further. Also, two other homothetic forms were estimated, but the above form was judged superior based on tests suggested in Ringstadt, Vidar, “Some Empirical Evidence of the Decreasing Scale Elasticity,” Econometrica, 42 (Jan. 1974), 87–101CrossRefGoogle Scholar. The maximum likelihood estimation procedure for the functions is discussed in footnote 28 below.

²⁷ An earlier version of this paper noted that the residual pattern for the Cobb-Douglas suggested some misspecification and that the homothetic form failed to provide a superior description. I am indebted to an anonymous referee for zeroing in on the residual problem and to Knox Lovell for pointing me to the correct estimation procedure which markedly changed the results.

²⁸ The equation is estimated using a maximum likelihood procedure outlined by Zellner, A. and Revankar, N. S., “Generalized Production Functions,” Review of Econmic Studies, 36 (Apr. 1969), pp. 241–50CrossRefGoogle Scholar. Rewriting (5) in logarithmic form gives us:

Under the assumptions that the u₁s are normally and independently distributed with zero mean and common variance σ², the logarithm of the likelihood function is:

where n is the sample size and J represents the Jacobian:

Substituting σ² for σ² in (iii) and rewriting J(θ, Y) reduces the log likelihood function to:

The maximization procedure entails two steps. For any given value of θ log 1^* is maximized by minimizing the second term in (v). This is done using ordinary least squares. From these estimates we can evaluate log 1^* – constant for each value of θ and thereby identify the global maximum of the function.

The estimate of θ() can be evaluated by noting that:

where η is the confidence level. Using η = 05 sets the right hand side of (vi) at 1.92. The log likelihood values generated in the estimation procedure can be examined to determine the range of θ for which the corresponding value is within 1.92 of the maximum. For my estimates the range did not include θ = 0 in either year; alternatively, I found that

and hence was able to reject the null hypothesis that θ equalled zero.

²⁹ Griliches, Zvi, “Specification Bias in Estimates of Production Functions,” Journal of Farm Economics, 39 (Feb. 1957), pp. 8–20CrossRefGoogle Scholar.

³⁰ Gallman, “Self-Sufficiency,” p. 14. Gallman's index increases with size. Inventories, however, decreased leaving meat per slave on farms with over 1,000 acres at two-thirds the level of farms with under 200 acres. As noted above, sugar farmers also purchased live swine for fattening. However, the measurement error would only equal the value added less feed already in the output figure. In regard to other crops, the assumption that only farms with over 500 acres produced these crops changed the scale estimate by only 02.

³¹ Vacuum includes any process using a vacuum pan for the strike, although some were supposedly better than others. The group includes the Rillieux apparatus for processing cane. This was one of the more significant technological advances of the 1840s, although it actually found little acceptance prior to the Civil War as claimed by other authors. This process linked the vacuum pans together so that exhaust steam from one could heat the next, thereby reducing fuel consumption. Rillieux, Norbert, “Rillieux's Sugar Machinery,” DeBow's Review, 5 (1948), pp. 291–93Google Scholar; Benjamin, Judah P., “Cultivation and Manufacture of Sugar,” DeBow's Review, 2 (1846)Google Scholar, reprinted in DeBow, J. B. D., ed., The Industrial Resources of the Southern States (New Orleans, 1856), III, 195–202Google Scholar.

³² The matched sample is also discussed in my dissertation and the appendix noted above. For an example of how the matched sample allows richer analysis than separate cross-sectional samples, see Donald Schaefer, “New Evidence on the Distribution of Agricultural Wealth in the Antebellum South,” manuscript, North Carolina A&T, 1976.

³³ The remainder is due to machinery productivity and may be due to a few farms with very low machine inputs. These farms probably did not process their own sugar.

³⁴ An additional impetus to growth came from changes in market conditions. While factor prices, especially for labor, rose during the decade, so did the price of sugar. The sugar price increase and productivity gains widened profit margins for most planters.

³⁵ A possible exception to my above argument that factor prices did not vary with size involves additional improved acreage added to cultivation. Levee building represented a large fixed cost incurred when the (arm was started, but newly cleared land did not involve additional levee protection. Hence, established farms had lower unit costs for extra land. This problem is probably insignificant since all frontage was. already taken by the 1840s, and all farms were facing equal additional costs, albeit lower ones.

³⁶ Gray, L. C., History of Agriculture in the Southern States to 1860 (Washington: Carnegie, 1932), II, 743Google Scholar. Sitterson, J. Carlyle, Sugar Country: The Cane Sugar Industry in the South, 1753–1950 (Lexington: University of Kentucky Press, 1933), pp. 138, 157–64Google Scholar.

³⁷ Whitten, David O., “Sugar Slavery: A Profitability Model for Slave Investments in the Antebellum Sugar Industry,” Louisiana Studies, 12 (Summer 1973), 426Google Scholar. This paper details the costs of various options available in the manufacturing stage and attempts to estimate profit rates for various combinations. The study is flawed, however, by its reliance on aggregate capital data for the atypical year of 1853.

³⁸ Maxwell, W. David, “Short-Run Returns to Scale and the Production of Services,” Southern Economic Journal, 32 (July 1965), pp. 1–14CrossRefGoogle Scholar.

³⁹ Mark Schmitz, Economic Analysis, p. 222. Additional farms traced across time showed similar experience. The twelve are cited merely as examples.