I wish to express my appreciation to Yoram Barzel and Douglass C. North who guided the dissertation upon which this article is based. Thanks are also due my colleagues M. Bruce Johnson, W. Douglas Morgan, John E. Pippenger, and Harold L. Votey, Jr. for helpful comments on earlier drafts of this article. Computations beyond those of the dissertation were done as part of a larger project supported by a grant from the National Science Foundation. Several suggestions from an anonymous referee resulted in significant improvement of the original manuscript.

¹ A useful collection of writings on the topic is contained in Carstensen, Vernon (ed.), The Public Lands (Madison: The University of Wisconsin Press, 1963)Google Scholar.

² The Central Pacific system was never legally known by that name. As used here, the Central Pacific system includes the Central and Southern Pacific railroads and all railroads leased and operated by them. See Table 1.

³ The others were the Union Pacific (including the Denver Pacific and Kansas Pacific); the Atchison, Topeka and Santa Fe; the Northern Pacific; and the Texas and Pacific Railroad systems. The five systems received about 75 percent of the land grant acreage. U.S. Federal Coordinator of Transportation, Public Aids to Transportation (Washington: U.S. Government Printing Office, 1938), II, 32Google Scholar.

⁴ Their position is summed up in the following statement by Senator William M. Gwin of California with respect to the operation of the pioneer railroads: “Our population would be increased, our resources doubled, and the continent covered with people and states from the Atlantic to the Pacific. Our wealth would be more than doubled; so would our products. A new impulse would be given to our agriculture, manufacturing, mining, commercial and navigating interest.” At the same time few doubted that these same railroads would fail, in fact would not be built for some period of time, without government aid.

⁵ One other estimate of rates of return for a land grant railroad has been published by an economist. Robert W. Fogel's study of the Union Pacific Railroad presented in The Union Pacific Railroad: A Case in Premature Enterprise (Baltimore: The Johns Hopkins Press, 1960)Google Scholar contains an estimate of the average ex post unaided private rate of return (11.6 percent) and the average ex post social rate of return (29.9 percent) for that railroad for the decade 1870–79. Fogel's estimates are clearly not applicable for testing the hypothesis that the land grant policy was beneficial with respect to other systems. Even if one wanted to accept them as proxies, they suffer from significant deficiencies. First, measurement of the economic contribution of the land grant subsidy to the firms involved is impossible since the private rate of return including the land grant aid is not estimated. Second, the social rate computation is affected by omissions of both benefits and costs. In the case of the pioneer railroad which reduces transport costs, external benefits accrue to three main groups: (1) non-railroad producers within the region served by the railroad; (2) shippers in interregional trade; and (3) passengers. Fogel includes only the first. A significant cost to society is also not included. This is the opportunity cost to society resulting from the federal construction loans (primarily) made to railroads in the Central and Union Pacific systems. This cost arises from the fact that the railroad firms were not required to (and did not) pay the full annual interest on the loans until (after) the loans fell due and at the same time did not pay interest on what was, in effect, a new loan—i.e., the unpaid interest. Fogel's estimated rates of return are computed by taking the average of the annual rates of return for the decade 1870–79 for the Union Pacific Railroad alone. An estimate of the relevant rates of return on the original investment in the Union Pacific and its affiliated roads—i.e., the Union Pacific system—would provide a better basis for judging the land grant policy's effect. The components of the railroad's capital stock—i.e., track, equipment, and structures—had lives of approximately 10, 20, and 50 years respectively. Construction of the Union Pacific Railroad began in 1865 and was completed in 1869. An approximation to the life of the original investment would be 1865 through the late 1880's rather than a decade in the middle of that life. Finally, Fogel used the ratio of the net earnings of capital to accumulated construction expenditures as the (unaided) annual private rate of return for the Union Pacific and the ratio of net earnings of capital plus annual external benefits to accumulated construction expenditures as the annual social rate of return. The (ex post) rate of return on an investment project is that rate of discount at which the present value of the net receipts stream (gross earnings or benefits appropriately defined minus the capital costs of the investment project) for the investment project over its life is zero. It is clear that the “ratio method” would only produce this rate of return by chance.

⁶ The equation PV = f(x) where is a polynomial with real coefficients and T roots. One difficulty with the internal rate of return method is that there may be more than one real positive root for which f(x) = zero. Descartes' rule of signs allows us to determine the maximum possible number of real positive roots and to state a necessary condition for there to be only one real positive root. The number of real positive roots is m − 2k where m is the number of variations of sign between successive terms of the polynomial written according to descending powers of the variable, and k is a positive integer or zero. The necessary condition for the possibility of only one real positive root for f(x), and, thus, the existence of only one internal rate of return, is that there be an odd number of sign variations. The stream of NR_t for each of the three rates of return to be computed here has five variations in sign between successive terms written in descending powers of the variable. Thus, the necessary condition is satisfied.

The PV curve for each of the three rates of return was computed by intervals of one percentage point for values of r from zero to 200. These curves demonstrate that for the PV functions used in this study only one real positive discount rate exists for which PV = O. In each case PV < O with r = 0 declines at an increasing rate (f″ (x) > 0) to a negative minimum as r increases and then increases at a decreasing rate (f″ (x) < 0) toward a negative asymptote as r increases toward infinity.

⁷ In real terms this is a measure of the reduced inputs per unit of output required for interregional shipments with the railroad. It is also a measure of the additional consumer surplus that results when an increased quantity of goods enters interregional trade at a lower cost.

⁸ The relevant demand curves are assumed to be linear.

⁹ It might be suggested that this is an underestimate of the true external benefits, which are actually (t*) (0Q″) + 1/2(t*) (Q″Q′). However, this is not the case. The point tQ″ was not observed before the railroad because D₂D₂ includes the effect of the extension of the zero rent perimeter, as transport costs fall with the railroad, and tQ″ is thus not relevant to the estimate of external benefits.

¹⁰ The only case in which the external benefits estimate made here would be the true stream of external benefits, given that demand increases, would be where the relevant demand curve was some curve D₃D₃ connecting the observed points before and after the railroad, as in Figure 3. It might be assumed that D₃D₃ is the long-run demand or supply curve for transport service, but it is not. It is a locus of equilibrium points rather than a demand or supply curve in the usual sense. However, the Observations which trace out D₃D₃ are used here to estimate the external benefits to interregional shippers.

¹¹ For detailed explanation, data, and sources concerning the construction and equipment cost estimate see Mercer, Lloyd J., “The Central Pacific System: An Estimate of Social and Private Rates of Return for a Land-Grant Aided Railroad System” (unpublished Ph.D. dissertation, University of Washington, 1967)Google Scholar, Appendix IIIA. For gross capital earnings see Appendix IIIC.

¹² Construction on the original Central Pacific line began in 1863 and was completed in 1869. Given the length of life of the various components of the capital stock cited above, a large fraction of the original capital stock of the system would have been wom out by 1889. Also, by 1889 the expansion and investment in the Central Pacific system by the original promoters had been completed. Thus, 1889 serves as an approximation to the end of the life of the original investment in the system.

¹³ All data used in calculating the rates of return in this study were computed in real or constant dollar terms. A number of price indexes were considered for this purpose. The Snyder-Tucker general price index, based on wholesale prices, cost-of-living, rents, and wages, appeared to be the best index on grounds of methodology and breadth of coverage. The base of the index was shifted to 1869 for use here.

¹⁴ Mercer, “The Central Pacific System,” Appendix IIIE.

¹⁵ The reasoning for adding the second item is as follows. The market value of the firm's common stock is the capitalization of the expected future residual gross capital earnings after all other claims have been paid. The government had a second mortgage claim on the future gross capital earnings stream. If the common stock market value (minus the value of future net land grant earnings) in 1889 was greater than zero (which it was), it was expected that the government claim would be paid out of the future gross earnings stream of the existing capital stock in 1889.

¹⁶ With this method the capital stock in a given year t is computed using the equation K_t = (K_t−1) (1 −δ) + (GI_t) (1 − δ/2) where,

The depreciation rate used was 3.69 percent. This was arrived at in the following manner. Based on the author's construction estimates, the three major components of the capital stock—(1) structures; (2) track; and (3) equipment—accounted for 65.9, 23.6, and 10.5 percent of the total, respectively, during the 1864 through 1869 construction period. On the basis of reported lives for these components, respective straight-line depreciation rates are 2, 10, and 5 percent. Track replacement was reported as a current operating expense and thus is not included in gross investment after the initial construction. Hence one can assume the track component is constantly maintained and not included in the depreciation rate to be charged against the capital stock. The weighted straight-line rate for structures and equipment is 1.843 percent—i.e., (.659)(.02) + (0)(.236) + (.105)(.05) = 0.01843. The double-declining balance rate is 3.69 percent and was used in computing annual capital stock.

¹⁷ For detailed explanation and data on the estimated net land-grant earnings, see Mercer, “The Central Pacific System,” Appendix IIID.

¹⁸ Macauley, Frederick R., Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856 (New York: National Bureau of Economic Research, 1938), pp. A37–A60Google Scholar.

¹⁹ Cowles, Alfred III, and Associates, Common-Stock Indexes, 1871–1937 (Bloomington, Indiana: Principia Press, Inc., 1938), p. 404Google Scholar.

²⁰ This is done using the approximation that where R_M = the money rate of interest, R_r = the real rate of interest, P = the prive level and ΔPe = the expected change in the price level:

The Snyder-Tucker price index (1869 = 100) was used to represent the price level. It was assumed that the change in the price level which actually occurred was expected in the preceding year.

²¹ Fogel, Robert W., Railroads and American Economic Growth: Essays in Economic History (Baltimore: The Johns Hopkins Press, 1964), pp. 75–79Google Scholar. Two modifications are made. First, no account is taken of the saving in hauling from water shipping points to primary markets, rather than from rail shipping points. Second, shipments to primary markets by water are not deducted from total shipments in estimating tonnage shipped per acre in farms. Since the extensive margin around a railroad rather than a water transportation facility is desired, the additional computations are irrelevant. For detailed information on estimation of the extensive margin in this case, see Mercer, “The Central Pacific System,” Appendix IVA. The second modification has been made since the writing of the work cited. Also a mistake in computing the cost per ton-mile for wagon haulage of the average tonnage shipped to primary markets per acre in farms has been corrected.

²² Fogel, The Union Pacific Railroad, pp. 100–1. Two implicit assumptions are made. One is that the nation's rate of population growth would have been unaffected by the absence of the Central Pacific system. This is likely since the increase in income due to the system was a negligible fraction of national income over the study period. Also, it is assumed that the rate of capital accumulation would have been unchanged in the absence of the Central Pacific system. If the rate of return on capital depends on factor proportions, and the rate of capital accumulation depends on the rate of return to capital, the absence of the Central Pacific system would have resulted in little change in the rate of capital accumulation. This is because the amount of supra-marginal land within the extensive margin around the Central Pacific system was a small proportion of the nation's supra-marginal land.

These considerations emphasize a significant difference between the present study and the well-known studies of Fogel and Fishlow, Albert, (Fogel, Robert, Railroads and American Economic Growth [Cambridge: Harvard University Press, 1965]Google Scholar, and Fishlow, Albert, American Railroads and the Transformation of the Ante-Bellum Economy [Cambridge: Harvard University Press, 1965])Google Scholar. Fogel and Fishlow took on the risks of applying the partial equilibrium methods of benefit cost analysis to a “large” project—i.e., the entire railroad system. Here only a relatively small subcomponent of the system is involved. While problems of partial equilibrium analysis are not absent in the present case, they are significantly reduced compared to the Fogel-Fishlow studies.

²³ Mercer, “The Central Pacific System,” Appendix IVB.

²⁴ Detailed information and explanation are presented in Mercer, “The Central Pacific System,” Appendix IVC.

²⁵ Details are given in Mercer, “The Central Pacific System,” Appendix IVD.

²⁶ This is certainly a conservative estimate and contributes to the downward bias in the estimated social rate of return.

²⁷ Several possibilities for placing the intraregional external benefits in the social returns stream are tested below. With a one-year lag, the estimated increment in land value between two census years is placed in the social returns stream one year after a county first fell within the extensive margin or the first census year as appropriate.

²⁸ Details of the estimate of the loan subsidy cost are presented in Mercer, “The Central Pacific System,” Appendix IVE.

²⁹ U.S. Federal Coordinator of Transportation, Public Aids, p. 60.

³⁰ Ibid., p. 59.

³¹ Thus for a county which was within the extensive margin by 1870, one-tenth of the estimated incremental pure land value between 1870 and 1880 is placed in each year from 1871 through 1880. A county which was first within the extensive margin between census years, for example in 1876, would be treated in the same way except that one-tenth of the 1870–1880 incremental land value would be placed in the social returns stream for 1877 through 1880. Thus only four-tenths of estimated incremental land value would be used in the social returns stream for the county in question. The underlying assumption here is that total “pure” land value in a county between census observations increases at a decreasing rate with respect to time—i.e., where PLV = total pure land value in county i, t = time.

³² The higher the absolute value of the net receipts stream, the higher is the value of r required to make the present value of that stream zero.

³³ The neglected indirect costs are the extra cost of cargo losses in water shipment, transhipping costs, additional inventory costs, neglected capital costs, and supplementary wagon haulage.

³⁴ Cf. McClelland, Peter D., “Railroads, American Growth, and the New Economic History: A Critique,” The Journal of Economic History, XXVIII (Mar. 1968), 102–23CrossRefGoogle Scholar.

³⁵ McClelland, “Railroads, American Growth,” p. 114.

³⁶ Provided one assumes unit costs would rise in the absence of the railroad. For the opposite conclusion, it would be necessary that unit costs fall for alternatives in the absence of the railroad. This does not appear to have been likely.

³⁷ Many writers in the past have argued that the land grants paid for the construction of the land grant railroads. Cf. Shannon, Fred A., America's Economic Growth (New York: The Macmillan Company, 1940), p. 363Google Scholar; and Morison, Samuel Eliot and Commager, Henry Steele, The Growth of the American Republic (New York: The Oxford University Press, 1942), II, 112–13Google Scholar. One basic fault with this is that systems rather than their components, i.e., individual railroads, are the relevant unit to consider. Moreover, the claim could only be true for individual railroads if we dealt with absolute dollar values and ignored the time dimension of the stream of construction costs and net land grant earnings. This too is certainly incorrect.

³⁸ It has been suggested that the acceleration of construction was ten to fifteen years at most. Cf. Riegel, Robert Edgar, The Story of Western Railroads (New York: The Macmillan Company, 1926), p. 43Google Scholar.