Duality Theory and Functional Forms for Dynamic Factor Demands

Abstract

A large body of empirical studies of factor demand systems (Christensen, Jorgenson and Lau (1973), Berndt and Christensen (1973), Fuss and McFadden (1978)) is based on the assumption of instantaneous adjustment by firms to prevailing prices. An exhaustive set of integrability conditions derived from the theory of static profit maximization is applied to guide the specification of functional forms and to motivate meaningful hypotheses to be tested or maintained. In contrast, such a sound theoretical basis is lacking for most of the empirical literature on dynamic factor demand (investment and employment) models. For example, Jorgenson (1965) appends an ad hoc lag structure to a theory of static profit maximization to generate an investment demand function. The recent developments in the adjustmentcost model of the firm (Lucas (1967), Treadway (1969), (1971), Mortensen (1973)) have provided a consistent dynamic theoretical framework for the determination of all inputs and outputs. Yet there still remains a gap between the theoretical model and most econometric factor demand models (Schramm (1970), Nadiri and Rosen (1969), Brechling (1975)); the latter maintain a flexible accelerator adjustment with constant adjustment coefficients while the theory implies that the flexible accelerator is generally optimal only locally in a neighbourhood of the steady state and that the adjustment coefficients generally depend on exogenous variables. (See Treadway (1974).) The studies cited also fail to relate satisfactorily the hypothesized adjustment matrices to the specified technologies. They thereby overlook some of the testable properties of the theory. Only Berndt, Fuss and Waverman (1977) have estimated a model that is fully consistent with the adjustment-cost theory. The objective of this paper is to describe a practical procedure for generating a large class of functional forms for dynamic factor demand functions that can be used to test and apply the adjustment-cost theory of the firm. Three limitations of the approach adopted by Berndt, Fuss and Waverman may be noted. First, their approach is practical only when there is a single quasi-fixed stock (or (footnote 6) perhaps at most when there are two quasi-fixed stocks). Second, the flexible accelerator is the only adjustment mechanism that can follow from their approach. Finally, they assume that firms expect current prices to persist indefinitely. In contrast the procedure developed below is applicable to any number of quasi-fixed stocks and is capable of generating a richer class of dynamic adjustment mechanisms, e.g. the hump-shaped lag distributions that have been estimated in the empirical literature on investment (Evans (1969; pp. 95-105)) and which have been modelled by various ad hoc lag structures such as the Almon, Pascal and rational distributed lags. However, the assumption of static expectations, common in the adjustment-cost literature, is maintained.