Appendix: The Sources of Economic Growth

APPENDIX: The Sources of Economic Growth

This appendix explains the derivation of the conclusions presented in Chapter II concerning the sources of economic growth for the period 1869–1914.

The basic assumptions are that the economy has an aggregate production function characterized by constant returns to scale and that all markets are in competitive equilibrium. Given these assumptions, the production function can be written as

(1)Y = A K^α L^1–α

where Y = total output, A = index of total input productivity, K = material capital stock (including land), L = man-hours worked, and α = share of property owners in total income. By taking logarithms and then differentiating with respect to time, we can express (1) as

(2)Y′= A′ + αK′ + (1–α)L′

where primes denote proportional rates of change. This states that the rate of growth of total output is the sum of the rates of growth of total input productivity, labor, and material capital, where the growth rates of the inputs are weighted by their relative income shares.

By subtracting L’ from each side of (2), we obtain

(3)y′ = A′ + αk′

where y = output per man-hour and k = ratio of material capital to labor. This states that the rate of growth of output per man-hour is the sum of the rate of growth of total input productivity and the weighted rate of growth of the material capital intensity, where the weight is the property share of income.

From data given in John W. Kendrick, Productivity Trends in the United States (Princeton, N. J.: Princeton University, 1961), pp. 298–99, 311–12, and 320–21, we obtain the following growth rates for the period 1869–1914: Y’ = 4.2 percent per year; L’ = 2.5 percent per year; and K’=3.9 percent per year. From data given in Edward C. Budd, “Factor Shares, 1850–1910,” in National Bureau of Economic Research, Conference on Research in Income and Wealth, Trends in the American Economy in the Nineteenth Century (Princeton, N. J.: Princeton University, 1960), p. 387, it appears that the relative income shares were approximately constant during the period at α = 1/3. Substituting these data in (2), A’ can be calculated, and therefore the proportional influence of three broad sources of growth in total output obtained as follows: the increase in man-hours accounts for 1.67/4.20 = 40 percent; the growth of material capital accounts for 1.30/4.20 = 31 percent: and all other sources explain 1.23/4.20 = 29 percent. Substituting the data into (3), we find that 0.46/1.70 = 27 percent of the growth in output per man-hour is explained by the increase in the ratio of material capital to labor, and 1.24/1.70 = 73 percent is attributable to all other sources.

These calculations are, at best, suitable only for establishing relative orders of magnitude. For full discussions of imperfections in the data, see the works by Kendrick and Budd cited above. For a penetrating survey of the theoretical reservations to which such calculations are subject, see M. Ishaq Nadiri, “Some Approaches to the Theory and Measurement of Total Factor Productivity: A Survey,” Journal of Economic Literature, VIII (Dec. 1970).