Factor-Price-Frontier Estimation of a "Vintage" Production Model of the Postwar United States Nonfarm Business Sector

Abstract

W l tE REPORT here on the construction and AV Vapplication of a method of estimating the parameters of an aggregative vintage model of potential output in the postwar United States nonfarm business sector. The model admits capital-embodied as well as unembodied technical progress. It posits that ex post and ex ante substitution possibilities between (of any vintage) and cooperating labor are alike. There are constant returns to scale. Ultimately, we resort for convenience to the Cobb-Douglas function, although certain estimates, such as that of the marginal productivity of investment, do not hinge upon this specification. Theoretical aspects of the model have been studied by Solow [11] and Phelps [7]. The model has been estimated by a conventional method most recently by Berglas [1] and Intriligator [5]. The principal novelty of our method of estimation is the use of Samuelson's factor price frontier construct [10]. This approach permits us to dispense with stock data. The virtue of this is that, frequently, stock estimates areunavailable because of the absence of early investment data. Moreover, as Fisher has recently proved [3], an aggregate production function relating aggregate output to aggregate effective capital exists, even in a one-product model, only if ex post and ex ante substitution possibilities are alike and any capital-embodied technical progress can be represented as solely capital-augmenting (i.e., as if all capital-embodied change could be expressed by a improvement factor). Since stock data for the United States business sector are available and since, in our model, there is ex post substitutability and embodied progress can be describ d as capital-augmenting, our method of estimation can be regarded as unnecessary. We present it, however, in the hope that other investigators will be stimulated to apply our method to more general models in which an aggregate production function does not exist and to countries for which no stock data exist. Further, we hope that our results will be welcomed for purposes of comparison with those from other techniques of estimating the same model. Turning to our results, we should like to emphasize that our statistical procedure is somewhat amateurish, that the data may be insufficiently accurate and that there is serious misspecification in the model (especially with respect to the ex post substitutability of and labor). Nevertheless, we cite the following findings for whatever they are worth: (1) the rate of technical progress increased significantly over the postwar period (accelerating technology); (2) the elasticity of output is much smaller than capital's relative share, which suggests that there is a large element of monopoly rent in capital's income; (3) embodied progress was negligible in the early postwar years but eventually exceeded unembodied progress by the end of the period; (4) the marginal productivity of investment began at about 30 per cent in the late forties, fell to 17 per cent in the middle fifties, and recovered to 25 per cent in the late fifties; (5) what we call the rate of obsolescence (the proportionate rate at which the price of new goods in efficiency units declines relative to the price of consumption goods) was about five per cent in the middle fifties and nine per cent in the late fifties; (6) so that, if the rate of physical depreciation was about four per cent, the one-period social rate of return was about eight per cent in the middle fifties and about 12 per cent in the late fifties. * This study was supported by the Cowles Foundation and the Economic Growth Center, both at Yale University. The work is also an exploratory effort in a joint project, Future United States Economic Growth, of M.I.T. and Yale. We are grateful to the members of this project for their suggestions and comments on this study.