The recent world energy crisis rekindled the interest of many economists in production functions and, especially in the substitution possibilities between energy and other inputs in production. Furthermore, the sharp increase in energy prices (relative to all other input prices) and the slow effect of this increase on the techniques of production, demonstrate the importance of properly incorporating short-run constraints into the characterization of a firm's behavior. The first purpose of this study is to establish a four factor production process for the industry level which will be consistent with the economic theory on the one hand, and will have the flexibility to incorporate short-run technological constraints on the other hand. This concept of the production process is used to derive the demand for factor inputs and the capacity utilization rate in a consistent manner. The second purpose is to use the model of the production process to analyze the possibilities of substitution among factor inputs (with special attention given to the energy input) and to investigate the role of capacity utilization in the demand for factor inputs. Given these objectives, we assume that the ex-ante production function with four inputs (capital, labor, energy and material imputs) can be approximated by a "Three Level CES Function" which takes the form: X = f{g(h(K,E),L),M} where each of the subfunctions f, g, and h are CES functions. This function does not impose identical or constant elasticities of substitution amone factors. We assume that the ex-ante behavior can be characterized by a cost minimization hypothesis. However, for the short-run behavior the hypothesis used in this study is a partly putty-clay approach, in which there is no direct ex-post substitution among factor inputs, and the installed capital embodies specific techniques with respect to labor, energy and material inputs. Furthermore, a specific capital stock is associated with a short-run full capacity output level embodied in that capital stock. This short-run capacity output can be produced when the installed capital is in full operation and all other inputs are provided to the production process according to the short-run techniques. Changes in the capacity utilization rate will result in changes in the demand for labor, energy, and material inputs, even though the production technique does not change. While old capital embodies old techniques, new capital is designed to suit the ex-ante optimal techniques in production. This relationship between the ex-ante techniques and the ex-post techniques in production is used to describe the path toward long-run steady state equilibrium. We tested our model on four different U.S. industries and demonstrate the power of our approach in explaining the demand for factor inputs. We estimated the full capacity output and capacity utilization rate in each period and show the significance of the capacity utilization rate in explaining the demand for factor inputs. Based on the estimated parameters, we compute the implied elasticities of substitution and show that in some industries capital and energy are substitute factors while in others the are complementary factors. We present an alternative model of the production function under the assumption of short-run profit maximizing behavior (under a fixed capital stock constraint) using the "putty-putty" case, in which there is no distinction between the ex-ante and the ex-post functions from the possibilities of substitution view point. We used this alternative model to estimate the demand equations for four U.S. industries under four alternative specifications. We compare the alternative approaches and show the superiority of the putty-clay approach in terms of the goodness-of-fit. Based on the empirical results, we adopt the putty-clay approach as a better approximation to reality.
