Nonpecuniary Rewards and the Aggregate Production Fuction

T has long been recognized that the provision of nonpecuniary rewards to a of production creates a corresponding reduction in the observed market price of the factor. But the effect of the provision of nonpecuniary rewards on marginal products and thus on the relationship between marginal products and observed prices, has long remained an open, albeit unpressing, question in economic theory. This neglected question has recently grown in importance as several influential papers concerned with the estimation of production functions (e.g., Solow 7 and ACMS 2) have been crucially based on the identification of marginal products with observed prices. Section I of this paper contains a generalization of the usual theory of the firm which allows for (1) joint production in a general form and (2) the existence of outputs which are not separately marketed. Differences between private marginal products and competitive prices, hereafter called discrepancies, are seen to be possible when and only when some of the firm's outputs are sold or evaluated in markets as nonpecuniary rewards. A factor's observed price is seen to equal its marginal product, its marginal product plus the reduction in payments to other factors made possible by a unit addition of the factor. Since the latter part of the net marginal product is not a derivative of an observed production function, the following empirical proposition becomes obvious: A factor's marginal product cannot be identified with its observed competitive price; nor can the magnitude or direction of the actual discrepancy be casually specified. This proposition immediately implies the existence of a fallacy, probably a crippling fallacy, in the modern approach to the estimation of production functions. Section II, which builds on the analysis of discrepancies in section I, contains the key result that any discrepancy can be considered a disaggregation biasthat when all firms are combined to form a competitively determined industry production function and all inputs are suitably grouped to form a single input index, the marginal product of the input index is equal to its average cost! To establish this, it is shown that for given relative output prices and given conventions on the indirect marketing of outputs, an ordinary industry production function generally exists in neoclassical, competitive equilibrium, that this aggregate production function is linearly homogeneous, and that observed payments exhaust the product; yet there are, in general, no equalities between prices and marginal products. This surprising result is used to show that the identification of the aggregate marginal product of a with its observed cost can be justified on neoclassical grounds only when the factor is a single index of all of the various factors of production. Thus, while the modern techniques of studying technology based upon identifying marginal products of individual factors with observed prices are theoretically groundless in the presence of nonpecuniary rewards, the techniques of testing for scale economies and measuring technical change introduced by CobbDouglas and by Abramovitz 1 and Kendrick 5 are valid applications of neoclassical theory under suitable treatments of the relevant input indices. Also rationalized is a technique devised by the present author 8 of estimating both the degree of aggregate returns to scale and the annual rate of technical change with a single index of all inputs.1 * This work was supported by the Institute of Government and Public Affairs at the University of California at Los Angeles and by the National Science Foundation under Grant G-16239. The author benefited substantially from a discussion with Karl Brunner and from the comments by a Referee on an earlier draft. 1 The empirical results of 8 strongly confirm a hypothesis suggested by the present paper; viz., that there is a great deal of inequality between the marginal products and prices of separate inputs but there is an equality between