We wish to thank Lou Cain, Robert Gallman, Stanley Engerman and De-Min Wu for their comments which have immeasurably improved the presentation. The remaining errors are our responsibility.

The necessary computer operations were carried out at the University of Kansas Computation Center. We acknowledge their assistance.

^{page no 854 note 1} Clapham, J. H., “Economic History as a Discipline,” in Lane, F. and Riemersma, J., editors, Enterprise and Secular Change (Homewood, 111.: Richard D. Irwin, 1953), p. 416Google Scholar. Emphasis added.

^{page no 854 note 2} Conrad, A. and Meyer, J., “The Economics of Slavery in the Ante Bellum South,” Journal of Political Economy, LXVI (April 1958), 95–122CrossRefGoogle Scholar. Also see Fishlow's review of Trends in the American Economy in The Journal of Economic History (1962), pp. 71–80, for a discussion of the desirability of improving upon single estimates.

³ The same analysis could also be performed for equal absolute changes in the two components.

⁴ Conrad and Meyer, “The Economics of Slavery…,” Table 9.

⁵ This assumption is a justification for the complexity of our subsequent case study. In some cases one might be able to generate a distribution from observed output data.

⁶ It would of course be necessary to know whether or not the decisionmaker had information similar to that which one is able to generate now.

⁷ Weibull, Waloddi, “A Statistical Distribution Function of Wide Applicability,” Journal of Applied Mechanics (Sept. 1951), pp. 293–97.Google Scholar

⁸ Weibull, p. 293. In this paper Weibull applied the distribution function to such diverse samples as Yield Strength of Steel, Length of Cyrtoideae, and Statures of male adults bora in the British Isles.

⁹ This asymptotic feature is useful in cost models as absolute upper bounds can rarely be set.

¹⁰ Conrad and Meyer, “The Economics of Slavery…, “Table 9.

¹¹ However, if the researcher has sufficient data for some variables and insufficient data for others, a mixed model can be used with probability distributions for some variables and best estimates for others.

¹² S. A. Sobel, “A Computerized Technique to Express Uncertainty in Advanced System Cost Estimates,” The Mitre Corporation (Sept. 1963).

¹³ This problem is discussed later in the paper.

¹⁴ The methodology underlying the computer program was largely derived from Lamb, W. D., “A Technique for Probability Assignment in Decision Analysis,” General Electric Major Appliance Division Technical Information Series, No. G7MAL02, MAL Library, Appliance Park, Louisville (March, 1967).Google Scholar The computer model used in this paper is an improved version of the Weibull model in Schaefer, D., Husic, F., and Gutowski, M., “A Monte Carlo Simulation Approach to Cost-Uncertainty Analysis,” RAC-TP-349, Research Analysis Corporation, McLean, Va. (March, 1969).Google Scholar

¹⁵ Lamb, “A Technique for Probability…, “p. 18.

¹⁶ If Mode 1 (M₁) is greater than Mode 2 (M₂) then the hypotheses might be:

The significance level chosen would probably be less stringent than is customarily used (.05) since the Weibull distribution is only an approximation. Each Monte Carlo output for M₂ is subtracted from the respective output from the M₁ run. If the proportion less than zero exceeds the significance level then H_o is accepted.

¹⁷ Lebergott, Stanley, “U. S. Transport Advance and Externalities,” The Journal of Economic History, XXVI, 4 (1966), 437CrossRefGoogle Scholar–65.

¹⁸ Another alternative would have been to have used probability distributions from some variables and best estimates for others. The exact cost models used and the data are presented in the appendix.

¹⁹ In the railroad model all probabilities are .10 except for the high value for variables 2, 3, and 10, and the low value for variable 3, all of which are .20. In the canal model, all probabilities are .10 except the high value for variable 2, which is .20.

²⁰ Canal capacity was obtained from an earlier work: Weiss, Thomas, “U. S. Transport Advance and Externalities: A Comment,” THE JOURNAL OF ECONOMIC HISTORY, XXVIII, 4 (Dec. 1968), 633Google Scholar.

²¹ Ibid.

²² We used a factor of 16.5, which is the quotient of the tonnage under consideration (4,320,000) divided by canal capacity (261,818 ton miles). This assumes divisibility of canals, an unreal assumption^ but the costs are such that use of a more realistic scalar of 17 would not alter the results.

²³ See the discussion in the appendices to Lebergott's article for evidence regarding the variation in factor proportions.

²⁴ This would be true because there would have been a limited sample upon which to make such a judgment. Also, in later years when more information was available a correlation between technical coefficients and outputs would not have been obvious. This is because the observed inputs for a given year would not correlate with differences in output because the inputs available were based on previous traffic volumes. If one could distinguish between stocks “and flows, and determine the appropriate time lag, then correlation could probably be determined. It is unlikely that the 1840 investor would have been so sophisticated, and we therefore assume he viewed these items as random variables.

²⁵ We could of course use best estimates for these coefficients, but in the present case chose not to.

²⁶ In fact, we use the same data here as are used in the ex post example.

²⁷ If we had assumed long run constant costs, then the railroad most-likely value would have been approximately $1200. The way in which we have constructed the situation in Figure 1 gives a test which biases the results against railroads at low traffic volumes. Our comparison in Figure 3 incorporates an assumption of constant returns to scale. Non-constant returns, however, may be closer to the view of the 1840 investor.

²⁸ Similar results hold for different levels of cost. Specifically, there is a smaller probability (.61) that the cost of carrying 261,818 tons one mile by rail will exceed the most likely cost of carrying an equal tonnage one mile by canal, than is the probability such would occur using the canal (.68).

²⁹ This makes it difficult to accept without reservation Taylor's, George Rogers generalization that “the advantages of the railroads were so great that even the strongest canals could not long retain a profitable share of the business.” The Transportation Revolution (New York: Harper and Row, 1968), p. 55Google Scholar.

³⁰ During the 1840's, approximately 400 miles of new canal were constructed, and in several places, existing canals were enlarged at costs exceeding the original investment. (G. R. Taylor, ibid., pp. 52–3) Railroads could have been built over these routes instead, but were not.

³¹ Fogel's work suggests that canals charged a lower price, but that was for later years when railroad competition had already had its impact on canal rates. In the earlier railroad years, around 1840, it is doubtful that canal investors envisioned that railroads would force them to lower rates.

³² These figures are obtained by deriving first a dollar per ton mile figure from Lebergott's evidence and then multiplying by 4,320,000 ton miles in the railroad case, and 261,818 ton miles in the canal case. (Lebergott, “U. S. Transport Advance …,” Table 1, p. 445.)

³³ Our most-likely estimates are based in part on Lebergott's evidence. However, we did make some changes, and used a great deal of additional information. Significantly, the remainder of our distributions are derived from additional data (some of which were not available to Lebergott) and it is the remainder of the distribution which confirms Lebergott's uncanny accuracy.

³⁴ At a lower traffic volume (that is, at canal capacity), and with the assumption that railroads operated under long run decreasing costs, the. hypothesis of no difference would be accepted at any reasonable level of significance. At this lower traffic volume the difference between railroad and canal costs is positive in 43 percent of the cases.

³⁵ When using probability techniques in an ex post sense these difficulties do not arise.

^{page no 881 note 1} This is for specified probabilities of .10.

^{page no 883 note 2} There is less change if the specified value is lower. For example, the probability that a value greater than $18,440 would be obtained at probabilities of .10 is 79 percent, while at .20 it is 82 percent. The value $18,440 is the most likely value of the distribution obtained with .10 probabilities on all cost components.

^{page no 883 note 3} We also tested the railroad model for sensitivity to error in these same variables, but at a lower level of traffic volume than was used in the calculations underlying Table 1. The results of this test were that the commonly used measures were changed significantly. When a higher level of uncertainty was placed on only one variable, the smallest change was a 9 percent increase in the most likely value. With increased uncertainty on both variables, all the commonly used measures changed in excess of 30 percent. The probability of obtaining a total cost in excess of a specified value (the most likely value obtained with .20 probabilities on both variables 2 and 10) changed from .365 to .520. This is more sensitive than at higher traffic volumes, but a more meaningful measure of sensitivity than the changes in the more commonly used measures. At low traffic volumes, all measures are more sensitive to error in variable 2, a fixed cost item, than is the case at high traffic volumes.

Similar calculations were made using the canal model, but only one variable, road investment per mile, was considered to be in error. The results of that test were, a 5 percent change in the most likely, a 15 percent change in the low and mean values, and a 30 percent change in the high value and in the range. The probability of exceeding some specified values changed by less than two percentage points.

^{page no 884 note 4} The high and low probabilities of .20, in contrast to .10.